Bull Math Biol DOI 10.1007/s11538-014-0055-3 ORIGINAL ARTICLE

Early Stages of Bone Fracture Healing: Formation of a Fibrin–Collagen Scaffold in the Fracture Hematoma L. F. Echeverri · M. A. Herrero · J. M. Lopez · G. Oleaga

Received: 28 July 2014 / Accepted: 10 December 2014 © Society for Mathematical Biology 2014

Abstract This work is concerned with the sequence of events taking place during the first stages of bone fracture healing, from bone breakup until the formation of early fibrous callus (EFC). The latter provides a scaffold over which subsequent remodeling processes will eventually result in successful bone repair. Specifically, some mathematical models are proposed to estimate the time required for (1) the formation immediately after fracture of a fibrin clot, described in terms of a phase transition in a polymerization process, and (2) the onset of EFC which is produced when fibroblasts arising from differentiation of chemotactically recruited mesenchymal stem cells remodel a previous fibrin clot by releasing a collagen matrix over it. An attempt has been made to keep models as simple as possible, so that a explicit dependence of the estimates obtained on relevant biochemical parameters involved is obtained.

L. F. Echeverri · M. A. Herrero (B) · G. Oleaga Departamento de Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid (UCM), Plaza de las Ciencias s/n, 28040 Madrid, Spain e-mail: [email protected] L. F. Echeverri e-mail: [email protected]; [email protected] G. Oleaga e-mail: [email protected] L. F. Echeverri Facultad de Ciencias Exactas y Naturales, Instituto de Matemáticas, Universidad de Antioquia, Calle 67 N 53-108, Medellín, Colombia J. M. Lopez Departamento de Morfología y Biología Celular, Facultad de Medicina, Universidad de Oviedo, Campus del Cristo B, Julian Claveria s/n, 33006 Oviedo, Spain e-mail: [email protected]

123

L. F. Echeverri et al.

Keywords Mathematical modeling of bone healing · Fibrin polymerization · Chemotaxis · Early callus formation · Moving differentiation fronts Mathematics Subject Classification

92C42 · 35K57 · 92C45

Abbreviations EFC CC MSCs BMP VEGF PDGF FGF TGF CT RT Tf TF

Early fibrous callus Cartilaginous callus Mesenchymal stem cells Bone morphogenetic proteins Vascular endothelial growth factor Platelet-derived growth factor Fibroblast growth factor Transforming growth factor Clotting time Retraction time Time to the formation of a fibrin clot Time to the formation of EFC

1 Introduction Bone repair is a robust physiological process, which is constantly acting in healthy individuals. In particular, bone microfractures induced by exercise (including walking) are routinely self-healed without even being noticed, although large microfracture condensation may result in severe injury (McKibbin 1978; Boyde 2003). Interestingly, bone self-repair makes use of key ingredients of the developmental program responsible for ossification during the embryonic stage (Ferguson et al. 1988, 1999). In fact, three partially overlapping stages can be distinguished after bone fracture occurs: initial inflammatory response, callus formation and bone remodeling. Of these, the inflammatory stage, which is absent in embryonic ossification, has received comparatively little attention, in spite of its fundamental role in the successful completion of the whole process, a fact that will be stressed below. In this work, we propose and discuss a simple model to analyze the early stages of bone fracture repair. These begin with bone breakup and extend up to the formation of the first elastic, fibrin-based scaffold (the so-called early fibrous callus, EFC). EFC provides a template that spans the whole fracture gap, and on which subsequent ossification and bone remodeling will take place. Failure in establishing such template prevents successful bone healing (Gaston and Simpson 2007). EFC appears in a comparatively short time: no more than 48 h in the case of thin induced fractures in rats (Gerstenfeld et al. 2003), a reference case for the studies herein conducted. Specifically, we shall pursue here the following goals: (i) to estimate the characteristic times of the main events leading to EFC formation and (ii) to ascertain their dependence on the kinetics of relevant underlying mechanisms as platelet activation, fibrin polymerization and fibrin clot remodeling mediated by fibroblasts. We will not be concerned

123

Early Stages of Bone Fracture Healing

here with mechanical effects. These will become particularly relevant at later stages, as EFC becomes a cartilaginous callus on which different remodeling processes will result in increasing stiffness of the unfolding scaffolds, which eventually result in functional bone formation (Gerstenfeld et al. 2003, 2006). More precisely, our plan is as follows. In Sect. 2 below, we review basic biological facts concerning the main events in fracture repair, particularly during the early stages to be considered here. We then discuss in Sect. 3 a mathematical model whose analysis allows us to estimate several characteristic times of the process under consideration. The first of them corresponds to the formation of a fibrin blood clot, the earliest structure generated during the healing process. In spite of its poor mechanical properties, such clot provides a first bridge to link fracture borders (Schmidt-Bleek et al. 2012). The generation of that clot is described in terms of a phase transition in a fibrin polymerization model. To analyze such phase transition, considerable insight is gained from the study of a particular asymptotic limit, namely the case where fibrin clotting is triggered by an instantaneous thrombin discharge. Under such assumption, our model can be explicitly solved, and the time for such clot to be formed is then estimated in terms of kinetic parameters of the various physiological processes (thrombin-induced activation, fibrin polymerization) involved. Once this fibrin network is in place, attention is paid to a first, and crucial, remodeling effect. This results from the release by fibroblasts of collagen over the fibrin net to give raise to a EFC. Such fibroblasts arise from asymmetric division over the fibrin clot of mesenchymal stem cells (MSCs) recruited from neighboring regions (periosteum). MSCs move inwards from the borders of the fracture gap under the influence of chemotactic factors released by activated platelets trapped within the fibrin clot (Street et al. 2000; Groothuis et al. 2010). The tip of the MSC front marks the transition from early fibrin clot (ahead) to EFC (behind). We show that MSC trajectories can be explicitly computed under a number of suitable, biologically reasonable assumptions. This allows us to obtain an explicit formula for the time to the formation of this EFC that, as in our previous case, depends on the kinetic parameters involved. Some illustrative values for these characteristic times are then provided in Sect. 4. Finally, a discussion on the results obtained as well as on possible future directions is gathered in Sect. 5. To keep the flow of the main arguments in the article, a number of technical issues related to the contents of Sects. 2 and 3 are, respectively, postponed to “Appendices 1 and 2” at the end of this article. We conclude this introduction by pointing out that mathematical models of different aspects of ossification and bone repair have received considerable attention in recent years (Komarova et al. 2003; Ayati et al. 2010; Graham et al. 2013; Fasano et al. 2010; Pivonka and Dunstan 2012); for later repair stages, see also Moreo et al. (2009), Moore et al. (2014), Prokharau et al. (2012). Some issues similar to those addressed in this work have been also discussed in related biological problems as wound healing, which shares a number of features with bone repair (Mc Dougall et al. 2006; Murphy et al. 2012; Javierre et al. 2009) in particular during the early stages here considered.

123

L. F. Echeverri et al.

2 Basic Facts on Bone Fracture Healing: The First 48 h When bone fracture occurs, damage is induced on a variety of tissues including surrounding muscular cells and periosteum as well as on bone itself. Bone is a highly vascularized tissue, and its fracture leads to many blood vessels across the fracture line being broken. Blood then runs away from disrupted vessels to the medullary canal between the fracture ends and rapidly coagulates to form a clot. On the other hand, immediately after fracture the resulting necrotic material induces an intense inflammatory response including widespread vasodilation, plasma exudation, edema and recruitment of acute inflammatory cells (Chung et al. 2006). As a consequence, a complex hematoma is generated that is of major importance for the beginning of bone healing, since it provides a first bridge between separated fragments of bone (Schmidt-Bleek et al. 2012). Such hematoma plays a very small mechanical role in immobilizing the fracture, but it serves as a fibrin scaffold and provides an adequate microenvironment over which repair cells may settle to perform their function. Formation of hematoma is a comparatively fast process. In the case of thin induced fractures in rat tibia, a case which will be considered here for comparison purposes, the fracture gap is about 2.5 mm in average, and hematoma can be observed by optical microscopy at about 3 h after fracture induction. See Fig. 1 below. A key constituent of hematoma is a fibrin net. Immediately after bone breaking, a coagulation process starts. Thrombin generated by blood vessel disruption induces the cleavage of fibrinogen present in blood to yield fibrin monomers, which quickly polymerize afterward. In this manner, a fibrin net is generated that in 6–8 h connects both sides of the fracture and provides a first scaffold on which bone will eventually be produced (Gerstenfeld et al. 2003; Einhorn 1998). Importantly, when fracture occurs, a number of growth factors (BMP, VEGF, PDGF,.. ) are released within the fracture gap (Bolander 1992). Such substances will mediate subsequent steps in fracture repair. As soon as a fibrin clot begins to form, new functional blood vessels as well as specific cell populations are recruited from the fracture boundaries. The cells involved directly in the repair of fractures are of mesenchymal origin and of a pluripotential nature. For definiteness, we shall refer to such precursors as mesenchymal stem cells

Fig. 1 Blood clot between the ends of a bone fracture. Bone tissue is stained in pink, whereas blood tissue has a brownish color. The immediate ends of the fracture border contain necrotic material (arrows). Picture taken from tibia from a 35-day old rat, 6 h after fracture (Color figure online)

123

Early Stages of Bone Fracture Healing Fig. 2 Early remodeling can be perceived in hematoma 24 h after fracture, with fibrillar material being found in some regions of the blood clot represented in pink (arrows). Bone depicted in brown (Color figure online)

Fig. 3 Image showing early invading cells derived from the cambium layer of the periosteum at the border of the fracture 24 h after fracture. Such cells show morphology of mesenchymal cells (arrows) (Color figure online)

(MSCs) in what follows. MSCs move in the wake of the new vascular bed created by incoming blood vessels. The latter can be seen that within 6–12 h near the fracture boundaries and between 24 and 48 h, they have invaded the whole fracture gap (Shapiro 2008). See Fig. 2 below where images taken 24 h after fracture induction are presented. MSCs are chemotactically recruited from the fracture borders by chemical cues released from activated platelets trapped in the fibrin clot (Hollinger et al. 2008) (see Fig. 3). MSCs may sustain either symmetric or asymmetric division (Malizos and Papatheodorou 2005). Symmetric division, whereby a MSC gives raise to two new MSCs, takes place whenever their initial number is not enough to keep the healing process on track. On the other hand, when a MSC reaches the fibrin net, it can undergo asymmetric division, thus giving raise to one MSC and one differentiated cell, which may be of various types: fibroblasts, chondrocytes or osteoblasts depending on microenvironment stimuli and on the stresses they are subject to Malizos and Papatheodorou (2005), Yoo and Johnstone (1998), Hughes et al. (1995). In particular, such lineages are known to depend on the density of the net on which division takes place (Colnot et al. 2006; Eghbali-Fatourechi et al. 2005; Rumi et al. 2005). Cell replication is stimulated by some of the growth factors present, whose action is in turn dependent on the

123

L. F. Echeverri et al. Fig. 4 Picture representing a EFC being formed 24 h after fracture. An increasing number of invading cells is observed (arrows), and partial reabsorption of the hematoma can be noticed (stars) (Color figure online)

unfolding of the fibrin clot. In particular, anchorage into the net significantly extends the half-life of growth factors with respect to that of freely diffusing ones (Sahni and Francis 2000; Yon et al. 2010). Specifically, the first type of asymmetric division takes place over a low-density fibrin clot and gives raise to fibroblasts. Once attached to the net, fibroblasts secrete a number of substances (collagens), which stick to the clot and change its mechanical properties (Eghbali-Fatourechi et al. 2005; Rumi et al. 2005). The resulting network, called early fibrous callus (EFC), is in place between 24 and 48 h after bone breakup (see Fig. 4). It has larger mechanical stability than its preceding fibrin net, although not enough to ensure working stability to the healing bone. Such stability will be achieved through subsequent remodeling processes. In particular, upon sticking to a collagen-enriched fibrin net, MSC can again undergo asymmetric division that now gives raise to one MSC and one chondrocyte (Malizos and Papatheodorou 2005; Colnot et al. 2006). This new type of cell interacts with the existing net and produces a more stable compound, called cartilaginous callus. Bone formation properly begins in a subsequent remodeling phase. At that stage, cartilaginous callus is replaced first by trabecular and then by lamellar bone (Takahara et al. 2004) through a process similar to endochondral ossification. These later stages will not be considered in this work.

3 A Mathematical Model for the Formation of a Fibrin–Collagen Callus Template In this section, we formulate and analyze a mathematical model to describe the early stages of cartilaginous callus formation in a fracture. For simplicity, we specialize here to a one-dimensional case and assume that gap is located at an interval   the fracture centered around the origin, say Ω = − L2 , L2 , with L > 0. Our model is of a modular character and contains a series of basic units that are activated in sequential order. These will be described below.

123

Early Stages of Bone Fracture Healing

A

B

Fig. 5 Typical thrombin profiles (adapted from Fasano et al. 2011; Butenas et al. 1999). a About 95 % of thrombin is generated as a consequence of positive feedback processes involving coagulation factors FVIII, FIX and FX (Fasano et al. 2011; Mann et al. 2003). b Dependence of thrombin concentration on time in some experimental settings, when anticoagulant factors are kept below hemostatic levels (Butenas et al. 1999)

3.1 Generation of Fibrin Monomers Upon Thrombin-Induced Fibrinogen Cleavage Right after a fracture is generated, a number of substances arising from broken blood vessels are allowed to enter in contact with leaked blood. The actual sequence of biochemical events thus triggered is known to involve several agents (Fasano et al. 2011; Orfeo et al. 2005). However, a common simplifying assumption consists in considering that complex process as driven by the action of the enzyme thrombin (obtained from its inactive precursor prothrombin), which cuts inactive fibrinogen molecules present in blood, thus giving raise to fibrin monomers. The latter then quickly polymerize to produce a fibrin clot. It is customary to describe thrombin generation by means of curves of the form represented in Fig. 5, below (Fasano et al. 2011; Butenas et al. 1999). As mentioned in the caption of Fig. 5, the sequence of events leading to thrombin generation include a number of feedback loops of an activator-inhibitor nature involving several coagulation/anticoagulation factors (Fasano et al. 2011; Guria et al. 2009, 2012). Given that large uncertainties remain about the order and rates of the corresponding chemical reactions (Danforth et al. 2009; Wagenwood et al. 2006), we have resorted here to a simplified modeling approach, which is now described. Let us, respectively, denote by F(t), θ (t) the concentration of fibrinogen and thrombin (assumed homogeneous in space) at time t. Following Guria et al. (2009), we postulate that fibrinogen is removed by thrombin according to a second-order kinetics (1) F˙ = −k g Fθ for some k g > 0, t > 0 whereas at t = 0, when fracture occurs, one has that F(0) = FM > 0

(2)

for some physiological concentration FM . We now consider an asymptotic limit of the profile sketched in Fig. 5 A, namely the case of an instantaneous thrombin discharge at t = TM > 0, that is:

123

L. F. Echeverri et al.

θ (t) = Cδ(t − TM ) for some C > 0

(3)

where δ(t − TM ) denotes Dirac’s delta at t = TM . Note that (1) and (3) represent a very fast chemical reaction for which the precise mathematical meaning of the governing differential equation has to be specified. In particular, (1) is to be understood here as stating: d (log F) = −k g θ dt On integrating this equation and using (2), it follows at once that:  FM for t < TM F(t) = FM e−kg C for t > TM

(4)

Fibrinogen molecules cleaved by thrombin give rise to fibrin monomers, which subsequently polymerize to form large fibrin aggregates. Using classical polymerization theory (Stockmayer 1943; Guy et al. 2007; Guria et al. 2009), the corresponding process can be modeled by an infinite system of coagulation–fragmentation equations for the concentrations Fk (t) of fibrin k-mers k ≥ 1 F˙1 = k g Fθ − F1

∞  i=1

a1i Fi +

∞ 

b1i Fi+1 − kr F1

(5)

i=1

∞ k−1 ∞   1 F˙k = am,k−m Fm Fk−m − Fk aki Fi + bki Fi+1 2 m=1 i=1 i=1 1  − bi j Fk − kr Fk ; k = 2, 3, . . . 2

(6)

i+ j=k

where ak j (respectively bk j ) denote the coagulation rates between k-mers and j-mers (respectively, the fragmentation rate at which k-mers are produced from the breakup of j-mers, j > k). Rates kr correspond to polymer deactivation and will be assumed to be independent of polymer length for simplicity. Assuming polymerization to occur according to classical Flory-Stockmayer theory, so that no polymer rings can be formed and coagulation rates are proportional to the number of the free available sites in each polymer, it turns out that ai j = k p (( f − 2)i + 2) (( f − 2) j + 2) (Guria et al. 2009; Guy et al. 2007) where f denotes the number of active functional sites in a fibrin monomer. Taking f = 4 (Wiltzius et al. 1982), we eventually obtain ai j = 4k p (i + 1)( j + 1)

(7)

On the other hand, assuming (i) the probability of a molecule of length k being cut by a fragmentation mechanism to be proportional to the number (k − 1) of bonds and

123

Early Stages of Bone Fracture Healing

(ii) the probability to form a molecule of length i out of fragmentation of a molecule of length k (k > i) to be inversely proportional to the number of bonds, we obtain (Guria et al. 2009) that: (8) bi j = 2kb for some kb > 0 A general discussion of the resulting system (5)–(8) is highly involved and will not be pursued here. However, considerable insight into blood-clotting dynamics is gained by making use of suitable auxiliary assumptions and simplifying hypotheses. To this end, for any integer n ≥ 0, we define the n-moment Mn of the polymer distribution: Mn (t) =

∞ 

k n Fk (t)

(9)

k=1

Note that M1 (t) represents the total mass of the polymer distribution once the monomer mass has been normalized to one. Of particular interest, here is the fact that the ratio (M2 /M1 ) corresponds to the average molecular size. In classical Flory–Stockmayer theory, a finite-time blowup of that ratio marks the onset of a sol-gel phase transition, whereby a gel phase appears (Stockmayer 1943). Bearing this fact in mind, we say that a fibrin clot is formed as soon as that ratio develops a singularity at some time (Guy et al. 2007; Guria et al. 2009). Upon replacing kb by 3kb , it follows from (5)–(8) that the moments M1 , M2 satisfy: M˙ 1 = k g Fθ − kr M1 M˙ 2 = k g Fθ + 4k p (M2 + M1 )2 − kb (M3 − M1 ) − kr M2

(10)

Note that such a system is not in closed form, since the third moment M3 appears in equation for M2 . To obtain a well posed problem, we now set: M3 =

M22 M1

(11)

As observed in Guria et al. (2009) where this closure formula was proposed, (10) holds true in two extreme cases, namely when either only monomers exist or when all fractions are concentrated in a single, large cluster and no smaller fractions are present. However, one always has that M22 ≤ M1 M3 by Holder’s inequality. Thus, when substituting (11) into (10), equality sign in the second equation for (10) has to be replaced by the corresponding inequality there. Therefore, the closed system M˙ 1 = k g Fθ − kr M1 M˙ 2 = k g Fθ + 4k p (M2 + M1 )2 − kb



M22 M1

 − M 1 − kr M 2

(12) (13)

is satisfied by solutions M1 to (12) and by subsolutions M2 to (13).

123

L. F. Echeverri et al.

3.2 Estimating Fibrin clot Formation As it turns out, Eqs. (12) and (13) can be explicitly solved under our current assumption (3). Notice that using (1), (12) can be rewritten in the form d  kr t e M1 = k g ekr t Fθ = −ekr t F˙ dt it then follows that M1 (t) = 0 for t < TM ;

 M1 (TM ) = FM 1 − e−kg C ≡ A

M1 (t) = Ae−kr (t−TM ) for t > TM

(14) (15)

Concerning the second moment, one also has that M2 (t) = 0 for t < TM ;

M2 (TM ) = A

(16)

Notice that the values M1 (TM ) = M2 (TM ) = A are obtained from the fact that by (1), (2) and (13), the jump of M1 , M2 at t = TM is provided by instantaneous activation of fibrinogen by thrombin at that time. We next point out that, on multiplying both sides in (13) by e−kr (t−TM ) and upon noticing that e−kr (t−TM ) = M1A(t) for t > TM (cf. 2 (t) (15)), one readily sees that U (t) = M M1 (t) satisfies: dU = 4k p Ae−kr (t−TM ) (U + 1)2 − kb (U 2 − 1) for t > TM dt U (TM ) = 1.

(17) (18)

Equation (17) is of Riccati type. Since U (t) = −1 is a particular solution, its general solution can be sought in the form U (t) = Y (t) − 1

(19)

To better understand the consequences of the various kinetic processes retained in (17), the following situations will be separately considered: 1. Case: k p > 0, kb = kr = 0. When fragmentation and deactivation are considered negligible with respect to coagulation, we obtain from (17) and (19) that U (t) =

1 + 8Ak p (t − TM ) 1 − 8Ak p (t − TM )

(20)

which blows up at a time T f given by T f = TM +

123

1 8Ak p

(21)

Early Stages of Bone Fracture Healing

2. Case: k p > 0, kb > 0, kr = 0. We now obtain from (17) and (19) that   kb + 4 Ak p 1 − e−2kb (t−TM )   U (t) = kb − 4 Ak p 1 − e−2kb (t−TM )

(22)

U (t) blows up at t = T f if 1 − e−2kb (T f −TM ) =

kb 4 Ak p

(23)

which requires kb < 4 Ak p

(24)

Therefore, large fragmentation rates preclude the formation of fibrin clots. When (23) holds, it follows from (23) that 

1 kb log 1 − T f = TM − 2kb 4 Ak p

(25)

Note that (25) reduces to (21) in the limit kb → 0. Moreover T f (kb ) is increasing in kb , so that (21) provides a lower bound for (25) when 0 < kb  1. The general situation where k p , kb and k g are all different from zero can be analyzed similarly to cases 1 and 2 above. However, for the ease of presentation, the corresponding details are postponed to “Appendix 2” at the end of this article. We conclude this section by observing that our discussion can be extended to the case where (3) is replaced by θ (t) = C(x)δ(t − TM ) for some function C(x). One just has to replace C by C(x) in the corresponding estimates, as for instance in (21). Note that this implies that T f = T f (x), so that different clot nucleation centers may be generated at different times. 3.3 Platelet Activation and Growth Factor Release Besides cleaving fibrinogen, thrombin also mediates platelet activation (Fasano et al. 2011). Activation leads to platelets changing shape, sticking to each other and releasing the content of their storage granules (Brass 2003). In the sequel, we will denote by pa (t) the concentration of activated platelets at time t > 0. Ignoring deactivation effects, we propose the dynamics of pa (t) to be given by: p˙ a (t) = α(P0 − pa (t))θ for t > 0 pa (0) = 0

(26) (27)

123

L. F. Echeverri et al.

Here, θ stands for thrombin concentration, P0 denotes the concentration of platelets at t = 0 and α > 0 is an activation rate. Recalling (3) and arguing as for (4), one obtains from (26) (27) that:  0 if t < TM pa (t) = (28) P0 (1 − e−αC ) if t > TM Upon activation, platelets are known to release a number of growth factors with mitotic and chemotactic effects, including members of the VEGF, FGF, PADG and TGF families (cf. Anitua et al. 2004; Cicha et al. 2004; Dimitriou et al. 2005; Rozman and Boita 2007). We next describe the time and space dynamics of one such factor g(x, t) irrespective of its specific biological function, which will be considered later on. L To that end, we consider our fracture gap Ω = ( −L 2 , 2 ) with L > 0 and assume that g(x, t) is released by activated platelets in Ω and freely diffuses inside and outside Ω. More precisely, we postulate that g satisfies: ∂g ∂2g − D 2 + βg = σ (t) pa (t)χΩ for − ∞ < x < ∞, t > 0 ∂t ∂x g(x, 0) = 0 for − ∞ < x < ∞

(29) (30)

where χΩ (x) = 1 when x lies in Ω, and χΩ (x) = 0 otherwise. Here, D and β denote, respectively, the diffusion and half-life coefficients of g(x, t). Concerning σ (t), we assume it to be of the form:  σ > 0 for TM < t < T ∗ (31) σ (t) = 0 otherwise for some constant σ and some T ∗ > 0. The reason for that choice is that platelets can only release such substances as they had initially stored, since they lack nuclei and hence the possibility to synthesize new molecules. The linear problem (29), (30) can be explicitly solved by classical methods to give: g(x, t) =

t 0

L/2

−L/2

 G(x − y, t − s)dy e−β(t−s) σ (s) pa (s)ds

where

(32)

x2

e− 4Dt G(x, t) = √ for − ∞ < x < ∞, t > 0. 4π Dt Recalling (28), for times TM < t < T ∗ (32) yields:  g (x, t) = σ P0 1 − e−αC

t TM



L/2

−L/2

 G (x − y, t − s) dy e−β(t−s) ds

(33)

(34)

whereas for t > T ∗ :  −αC g (x, t) = σ P0 1 − e

T∗ TM

123



L/2 −L/2



G (x − y, t − s) dy e−β(t−s) ds

(35)

Early Stages of Bone Fracture Healing

3.4 Recruitment of Mesenchymal Cells (MSC): Formation of Early Fibrin–Collagen Callus (EFC) Having described the dynamics of a generic growth factor g (x, t) in our previous Section, we now analyze the chemotactic effects induced by growth factors in the population of MSCs and their impact on remodeling the fibrin net to produce a first fibrin–collagen template. We assume that a baseline population of MSC is located at the periosteum, near the fracture border, and that when stimulated by g (x, t) such population moves inwards into the fracture gap. When doing so, MSCs eventually become precursors of various cell types instrumental to achieve bone repair. More precisely, let us denote by m (x, t) the concentration of MSCs at a point x at a time t. We postulate that m satisfies:

 ∂g μm = 0 ; −∞ < x < ∞, t > TM ∂x L L for − ≤ x ≤ , m (x, TM ) = 0 2 2 L L m (x, TM ) = m 0 for x ≤ − , x ≥ . 2 2 ∂ ∂m + ∂t ∂x

(36) (37) (38)

where μ > 0 is a chemotactic coefficient and m 0 is the concentration of MSC available at the periosteum reservoir. Notice that both diffusive and mitotic effects have been ignored in (36). Using (34), (35), an explicit formula can be obtained for ∂∂gx . To this end, we note that on setting: 2 erf(u) = √ π



u

e−x dx 2

(39)

0

one has that

(x−y)2

e− 4Dt I (x, t) = dy √ 4π Dt −L/2      Tc Tc 1 erf = (1/2 + x/L) + erf (1/2 − x/L) 2 t t L/2

where Tc =

L2 . 4D

(40)

(41)

Then, there holds: ∂I 1 (x, t) = √ ∂x π

Tc Tc 1  − Tc (1/2+x/L)2 2 e t − e− t (1/2−x/L) . t L

123

L. F. Echeverri et al.

Fig. 6 The form of ∂∂gx for t¯ = 0.01. All parameters involved are set equal to one for convenience

  Setting Pˆ0 = σ P0 1 − e−αC it follows that for TM < t < T ∗ : ∂g (x, t) = Pˆ0 ∂x



∂I (x, t − s) e−β(t−s) ds ∂ x TM t Tc Tc 2 Pˆ0 Tc  − 4(t−s) (1+2x/L)2 e = √ − e− 4(t−s) (1−2x/L) e−β(t−s) ds . π L TM t − s (42) t

At this juncture, it is convenient to make use of nondimensional variables given by: u=

t −s t − TM 2x , τ= , t¯ = L Tc Tc

(43)

∗

M Upon introducing (43) for times t¯ such that 0 < t¯ < T −T (42) can be recast in the Tc form:     2 2 e−Tc βτ Pˆ0 Tc t¯ − 21+u ∂g   √ √ − 1−u τ u, t¯ = √ −e 2 τ (44) e √ dτ ∂x πL 0 τ

When t¯  1 so is τ , and the integrand in (44) is almost zero except for values  close  to interval boundaries u = ±1. For instance, when t¯ = 0.01 a plot for ∂∂gx u, t¯ is provided in Fig. 6 below. When t¯ = O (1), that is for times t such that t − TM = O (Tc ), the profile for ∂g  ¯ ∂ x u, t looks as depicted in Fig. 7. Such a profile is maintained until it starts to decrease for t > T ∗ . Therefore, we will focus on the values of ∂∂gx for the time interval

123

Early Stages of Bone Fracture Healing

Fig. 7 A plot of ∂∂gx for t¯ = 1. All parameters involved are set equal to one for convenience

TM < t < T ∗ . The main contributions to the integral defining close to the boundaries u = ±1

∂g ∂x

in (44) are located

 −Tc βτ

 1 Pˆ0 Tc t¯ e ∂g  1 − exp − −1, t¯ = √ √ dτ ∂x τ πL 0 τ

(45)

  one may obtain a plot for ∂∂gx −1, t¯ as a function of t¯ that looks as in Fig. 8:   Note that ∂∂gx −1, t¯ quickly saturates to a constant value. Actually, one has that: t¯  ∞ e−Tc βτ e−v Tc β 1 1 1 − e− τ 1 − e− v √ dv √ dτ ≤ √ τ v Tc β 0 0

∞ −v e π 1 ≤ √ √ dv = Tc β v Tc β 0 Hence, μ

 ∂g  −1, t¯ ≤ μ Pˆ0 ∂x

√

Tc /β ≡ μa , L

(46)

and a similar result can be derived at u = 1. We are now ready to estimate the time it takes for a front of MSCs to reach the center of the gap at x = 0. Such front will be defined as the characteristic curve x(t) of (36) that starts from x = −L/2 at t = TM .

123

L. F. Echeverri et al.

  Fig. 8 A plot of ∂∂gx −1, t¯ as a function of t¯, showing quick stabilization to a plateau-like profile. Parameter values as in Figs. 6 and 7

A similar argument applies for the case where x(TM ) = L/2. Clearly, x(t) satisfies : ∂g dx =μ dt ∂x x (TM ) = −L/2

(47)

Once x(t) has entered into a linear regime (which in particular occurs for times t − TM ≈ 3TC , see Fig. 8), we have that ∂∂gx ≈ −2ax/L for 0 < T ∗ − TM /TC and L/2 < x < 0, and the differential equation in (47) can be approximated by : 2μa dx =− x dt L whence



L − x (t) = − e 2

2μa L (t−TM )

(48)



(49)

We may now estimate the time TF of formation of EFC as follows. Due to the approximations made to derive (49), the function x(t) defined there will never reach x = 0 in a finite time. We will thus say that EFC has been formed when x(t) reaches instead a value −ε with 0 < ε  1 arbitrarily selected. We then may compute TF from the relation:

123

Early Stages of Bone Fracture Healing

L 2μa − e− L (TF −TM ) = −ε 2 which gives at once:

√ TF = TM +

 β DL L , log ˆ 2ε μ P0

(50)

provided that T ∗ > TF (see (44)). 4 A Sequence of Times We now briefly discuss on the comparative values of the characteristic times obtained in our previous sections, as well as on their relevance in terms of blood clot formation. To begin with, we observe that a correct timing of the overall process requires that TM < T f < Tc < TF < T ∗

(51)

where TM , T f , Tc , TF and T ∗ have been, respectively, defined in (3), (21), (41), (50) and (31). For simplicity, we take TM = 0 in the sequel. On the other hand, it is reasonable to select T ∗ of the order of the average platelet life span, so that we may take: (52) T ∗ ≈ 10 days (cf. Kameth et al. 2001). To estimate the values of the remaining times in (51) is in principle feasible by means of the corresponding formulas obtained. However, while some of the parameters appearing there can be reasonably guessed (at least from in vitro experiments), little can be said about parameters as α and σ appearing in (26) and (29), respectively. We next remark on the dependence of the sequence of times in (51) on the kinetic parameters involved, and provide some preliminary estimates on their order of magnitude. To begin with, consider the value obtained in (21) for T f , which corresponds to the earliest stages of fibrin clot formation when fragmentation and deactivation effects are ignored. As to the parameters appearing in (21), of these, k p (respectively, k g ) represents a kinetic rate corresponding to fibrin polymers coagulation (respectively, to fibrinogen activation by thrombin) and FM can be considered as a homeostatic parameter (average fibrinogen concentration). The thrombin discharge C resulting from blood vessels disruption is in principle dependent on the nature of the induced fracture event and may be assumed to have a universal form for fractures generated under similar conditions (as, for instance, those described in “Appendix 1”). A quick glance at (21) reveals that the dependence of T f on FM and k p is of the form T f ∼ F1M , T f ∼ k1p . In particular, all other parameters remaining equal, diminishing the efficiency coagulation factor k p to a half (e.g., by adding an anticoagulant drug) results in doubling T f . The dependence of T f on k g and C is accounted for by the term 1 − e−kg C in (21) where both parameters jointly appear in a exponential factor. Clearly, T f goes to zero (respectively, to infinity ) as k g C goes to infinity (respectively, to zero). For small values of the product k g C, one has from (21) that T f ∼ O( kg1C ).

123

L. F. Echeverri et al.

In particular, the times T f corresponding, respectively, to values k g C = 0.1 and k g C = 0.3 change approximately by a factor of 0.3. The expression for T f in (21) can be compared with the prothrombin time T p , a parameter routinely measured in blood tests (Besselaaar 1996). T p is used to estimate the efficiency of the initial thrombin burst resulting from activation of the extrinsic coagulation pathway and is measured in decalcified plasma where platelets have been removed. A typical value obtained is T p ≈ 10 s. If we compare the fastest estimate for T f obtained in Sect. 3.2 (see (21)), we should have: Tf =

1   ≈ 10 s 8FM k p 1 − e−kg C

(53)

FM in (53) is commonly estimated in blood tests. A typical value for FM is FM ≈ 3 g/l = 9,000 nM

(54)

(see Guria et al. 2010). Fibrin polymerization rates seem to have been reported only in vitro (Guria et al. 2010 and references therein). For instance, we may take k p = 4 × 10−3 (nM · min)−1

(55)

(cf. Weisel et al. 1993). One then readily sees that 8FM k p ≈ 4.8 (s)−1 . Then, estimate (53) follows provided that 1 (56) 1 − e−kg C = 48 which holds provided that k g C ≈ 0, 0211. As a matter of fact, values for k g have been reported (Weisel et al. 1993; Guria et al. 2010). For instance, it has been suggested that (57) k g = 3 × 10−4 (nM · min)−1 (cf. Weisel et al. 1993). From (56) and (57), we obtain that under such assumptions C≈

104 FM 0.0211 nM · min = 0.47FM s ≈ s. 3 2

(58)

Interestingly, the value obtained for the thrombin discharge in (58) is compatible to that obtained in Butenas et al. (1999) for its inactive precursor prothrombin, which is about 79 FM . However, it has to be stressed that, as remarked in Sect. 2, the time to develop a mature fibrin clot is considerably larger than the value provided in (53). This is a consequence of the sequential activation of the whole set of activation-inhibition modules that eventually integrate into the whole coagulation cascade. In particular, a number of fibrinolytic processes quickly develop to counteract fibrin clotting, as for instance the fragmentation mechanism represented by the third term on the right of (13) (Fasano et al. 2011; Guria et al. 2012). To keep track of the actual unfolding of a fibrin net, quantities other than the prothrombin time are measured in clinical practice Rodak et al. (2011). Examples

123

Early Stages of Bone Fracture Healing

are the clotting time, CT, defined as the time required for blood to form a initial clot from a given volume of blood in a glass tube, which usually ranges between 5 and 15 minutes, and the retraction time RT, defined as the time taken for a blood sample to become solid enough, so that it could separate from the walls of a crystal vessel where coagulation is taking place and that ranges between 1 and 2 h. As a matter of fact, times as those reported for CT or RT can be obtained from our formulas (21) or (25) by selecting C and kb in a suitable manner there. It is worth remarking here that if we assume values of a thrombin discharge of the order of (58), we obtain a value for T f similar to that corresponding to RT by selecting kb in (25) close to the limit of validity of that formula, which is given by kb = 4 Ak p . Note in this respect that parameters as k p and kb are likely to have been selected in the course of evolution to remain confined within narrow values, as is the case for parameters in virtually all homeostatic processes analyzed so far (Ebenhoh and Heinrich 2001). Consider now Tc given in (41) through the relation L2 (59) 4D where D is the diffusion coefficient of the growth factor considered. Since not much seems to be known about in vivo values for D, we may now try on (59) a typical diffusivity value, say D = 10−6 cm2 /s. Taking now L = 2 mm (see “Appendix 1”), we obtain: Tc =

Tc = 104 s ≈ 3 h

(60)

Notice that by (53) and (60), T f  Tc . Moreover, a slower diffusivity will result in an increased value of Tc according to (59). Let us now remark on TF , the time required for early callus formation, for which −N we have obtained estimate (50) in Sect. 3.4. For definiteness, let us take ε = e 2 for some integer N there. Or taking N such that N log(L), we see that √

TF ≈

β DL N   μσ P0 1 − e−αC

(61)

Some of the parameters in (61) are known within reasonable ranges. For instance, values for half-lives of growth factors have been reported in the literature, although they considerably differ from one reference to another. For definiteness, we take here β = 6 · 10−4 (min)−1 (Beenken and Mohammadi 2009). Keeping to our previous choices and D = 10−6 cm2 /s, and selecting N large enough, we deduce √ L = 2 mm −4 that β DL N ≈ 10 cm2 /s. Platelet counts are commonly made in blood tests, with values ranging between 1.5·105 to 40·105 units per cubic millimeter, and mean platelet weight has been estimated to be about 10 picograms (Kiem et al. 1979). We therefore may take P0 = 2g/l as a reference value. On the other hand, little seems to be known about in vivo chemotactic parameters for growth factors. See however (Tranquillo et al. 1988), where μ has been measured for FNLLP-estimulated neutrophils , as well as (Stefanini et al. 2008) where data for free and bound VEGF are reported. Bearing these results in mind , it makes sense to consider values for μ so that μP0 = 10−5 cm2 /s. In this manner, we will obtain:

123

L. F. Echeverri et al.

√ β DL N

10 μP0

(62)

  Concerning the remaining parameters in (61), it is natural to assume σ = O T1∗ where T ∗ is given in (52). On taking σ = T1∗ and selecting then the conversion factor 1 , we would eventually arrive at 1 − e−αC 50 TF

T∗

2 days 5

(63)

Needless to say, our previous estimates have to be considered as very preliminary. For instance, for times of the order of CT or RT, additional characteristic times of the various coagulation and fibrinolytic networks involved need to be investigated. Actually, CT is used as part of hemophilia test, so that the dynamics of individual coagulation factors as FVIII need to be accounted for there. On the other hand, once the fibrin clot has been formed (which begins to occur quite fast; see Bolander 1992), the half-life of growth factors can be extended due to their anchorage in such a net (Sahni and Francis 2000; Yon et al. 2010). According to (61), an increase in that value from β to 4β would result in doubling the corresponding value in (61). This effect, however, can be somehow compensated by the slowdown in diffusivity D as the density of the medium increases. 5 Concluding Remarks In this work, we have used mathematical models to explore the early stages of bone fracture repair. It should be noticed that human bone has an impressive capability for self-healing. In fact, the human skeleton is constantly being renewed, and bone microfractures induced by exercise are routinely self-repaired (without being even noticed) by means of a microscopic mechanism termed as bone remodeling (Boyde 2003). It has been observed that bone remodeling makes use of some ingredients of the developmental program that drives ossification during the embryonic stage (Ferguson et al. 1999, 1988). However, as soon as fractures reach a millimeter-width size (as those considered in “Appendix 1”), inflammatory and coagulation processes appear which were absent at the embryonic stage, but play now a key role in stabilizing the fracture. The analysis of such early stages by means of suitable mathematical models has been the goal of this work. When faced with the goal of deriving mathematical models for bone fracture repair, a significant obstacle has to be addressed, which is also found in many other biological problems. More precisely, while many of the key agents in the process under consideration have been identified, little is known concerning the precise quantitative manner, whereby these agents coordinate themselves to achieve bone repair. For instance, only partial information seems to be available on the detailed network hierarchy of the corresponding biochemical cascades, and precise values for the order and rate constants of the chemical reactions involved in cell signaling during that process remain largely unknown. For these reasons, such parameters are often selected ad hoc to fit observed

123

Early Stages of Bone Fracture Healing

behaviors by means of model simulations. However, it is well known that a given set of experimental values can be reproduced by simulating different models with different sets of parameter choices, so that the insight gained through extensive parameter fitting has to be carefully assessed (Hemker et al. 2012; Ginzburg and Jensen 2004). Bearing these remarks in mind, we have aimed in this work at discussing simple models that provide relevant information on the specific issues addressed, namely the timing of events taking place at the early stages of bone fracture repair, and on their dependence on the biochemical processes involved. Specifically, we restrained our attention to the period lasting from bone breakup until a first fibrous callus (EFC) is formed. Successful and timely formation of EFC is crucial for fracture repair, since EFC is formed immediately after trauma, and provides the first template over which bone will be eventually produced by a sequence of remodeling processes. EFC formation is comparatively simple compared with later stages (for instance , mechanical effects do not yet play so relevant a role as in posterior remodeling). However, it already displays some features that will play a key role in subsequent bone repair, as will be recalled below. In Sect. 3, we first discussed how a fibrin clot, the earliest structure needed for successful fracture healing, unfolds after bone fracture. The onset of such a fibrin network is described here in terms of a sol-gel phase transition, and explicit formulas for the time required for the formation of such clot are derived under the assumption of an instantaneous thrombin burst (see Sect. 3.2 and subsequent discussion in Sect. 4) . We then analyzed the manner in which MSCs (mesenchymal stem cells) are chemotactically recruited by growth factors released from the early fibrin clot. When reaching that target, MSCs will divide and differentiate into fibroblasts, which upon sticking to that clot will in turn release a collagen extracellular matrix to remodel the former fibrin clot into a EFC. The covering of the fracture gap by a moving front of MSCs will be repeated in later stages, since the new lineages required (e.g., chondrocytes) for subsequent remodeling are also produced over previously extant scaffolds. Thus, the situation discussed in Sects. 3 and 4 provides a first step toward further work on the remaining stages of bone repair. Concerning the simplicity of our models, a few remarks are in order. When dealing with any of the processes considered, we have focused on the main features needed to derive the sought-for information, namely the times taken for relevant structures to develop and their dependence on the parameters involved. To do that, we have considered effective processes, so that various biochemical cascades have been lumped into a single equation, rather than exploring the intricacies of each aspect involved. Moreover, we have used as a starting point the assumption that thrombin is instantaneously discharged after bone fracture, a hypothesis in line with known facts about thrombin activation (Boyde 2003; Fasano et al. 2010). In particular, we have shown that a thrombin burst as that given in (58) triggers fibrin polymerization, so that a fibrin clot is generated in due time. More precisely, we have estimated an early fibrin time T f , which marks the initiation of clot formation and is comparable to prothrombin times T p measured in clinical tests (see 53). However, formation of a stable fibrin network takes longer than times of order T p . For instance, it is known that other (longer) time scales measured for instance by clotting times (CT) and retraction times (RT) have to be considered for that purpose. We have discussed such issue in Sect. 4. In particular,

123

L. F. Echeverri et al.

we have pointed out there that times of the order of the RT can be derived from formulas as (25) when fragmentation effects are considered, and fragmentation coefficients are close to the limit of validity of that formula. We have then discussed in Sect. 4 how times for EFC formation can be retrieved from the motion of MSCs fronts (50). Such times are shown to be comparable with those recalled in Sect. 2 for the experimental case described in “Appendix 1”, provided that reasonable assumptions (consistent with previously reported values) are made in the kinetic parameters appearing in (50). It has been already remarked that the form of the estimates obtained allows to examine the impact of strategies aimed at accelerating (or slowing) the underlying processes (cf. the discussion of the dependence of T f on coagulation and fragmentation parameters in Sect. 4). However, additional analysis is required to improve the estimates already obtained and to measure some of the parameters on which such estimates depend and that are likely to be of a structural, universal character that has conceivably been arrived at in the course of evolution. For instance, chemotactic parameters for the motion of MSCs under suitable chemical cues might in principle be obtained similarly to what was done in Tranquillo et al. (1988) for leukocyte motion. In addition to this, other aspects that have not been addressed here will deserve attention. A particularly relevant example of the latter is the evaluation of the robustness and redundancy properties induced by the simultaneous action of several growth factors with partially overlapping functions. We intend to address this and other related issues in future work. Acknowledgments L.F.E wants to thanks Colciencias and University of Antioquia for their support during the preparation of this work. M.A.H. and G.O. have been partially supported by MINECO Grant MTM2011-22656.

Appendix 1: Experimental Procedure Sixteen ten-week-old male Sprague-Dawley rats were obtained from the animal facility building of the University of Oviedo. Bone fractures with 2.5 mm average width were produced by the manual breakage using plate-bending devices, placed at the distal-third of the right tibia. The animal study protocol was approved by the local committee and conformed to Spanish animal protection laws. Rats were anesthetized and sacrificed by cervical dislocation 3, 6, 12 and 24 h after fracture (n = 5). Tibiae were isolated and cut through the sagittal plane of the metaphyseal fracture region into two equal sized parts and fixed by immersion in 4 % paraformaldehyde at 4 ◦ C for 12 h, rinsed in PBS, decalcified in 10 % EDTA (pH 7.0) for 48 h at 4 ◦ C, dehydrated through a graded ethanol series, cleared in xylene and embedded in paraffin. Sections were serially cut at a thickness of 5 m, mounted on Superfrost Plus slides (Menzel-Glaser), stained with a trichromic method using Weigerts hematoxylin/alcian blue/picrofuchsin and viewed and photographed on a Nikon Eclipse E400 microscope. Appendix 2: Technical Details We gather here a number of technical details related to issues that have been mentioned in the main text but were skipped there.

123

Early Stages of Bone Fracture Healing

A General Case in Blood Clot Formation Case: k p > 0, kb > 0, kr > 0. When all kinetic rates are different from zero substitution of (19) into (17) leads to dY = f (t)Y 2 + 2kb Y ; dt

f (t) = 4k p Ae−kr (t−TM ) − kb

(64)

so that Z = − Y1 solves dZ + 2kb Z = f (t), dt whence

Z (TM ) = −

1 2

 t 1 Z (t) =e−2kb (t−TM ) − + e2kb (s−TM ) f (s)ds 2 TM  1 4 Ak p −kr (t−TM ) e − e−2kb (t−TM ) − = 2kb − kr 2

(65)

for t > TM when 2kb = kr , and: Z (t) = 4 Ak p (t − TM )e−2kb (t−TM ) −

1 2

(66)

for t > TM and 2kb = kr . Assume now that 2kb > kr then, on setting:

it follows that U (t) =

Q=

4 Ak p 4FM (1 − e−kg C )k p = 2kb − kr 2kb − kr

M2 (t) M1 (t)

is given by:

  1 + Q e−kr (t−TM ) − e−2kb (t−TM )   U (t) = 1 − 2Q e−kr (t−TM ) − e−2kb (t−TM )

(67)

(68)

Let g(t) = e−kr (t−TM ) − e−2kb (t−TM ) for t ≥ TM . Clearly g(TM ) = 0, g(t) > 0 for t > TM and g(t) achieves a maximum at some time t = t ∗ . A straightforward computation yields that: 

1 2kb log 2kb − kr kr − kr  2kb



2kb −kr 2kb 2kb − 2kb −kr ∗ ∗ m = g(t ) = − kr kr t ∗ = TM +

(69) (70)

Thus, for U (t) in (68) to blow up at time T f , we need 2Qm ∗ ≥ 1

(71)

123

L. F. Echeverri et al.

where Q, m ∗ are respectively given in (67) and (70). If (71) holds, one has that 

1 2kb log T f ≤ TM + 2kb − kr kr

(72)

the case where 2kb < kr is similarly dealt with. Finally, when 2kb = kr blow up for U (t) occurs if there exists T f > TM such that 8Ak p (T f − TM )e−2k p (T f −TM ) = 1

(73)

  It is readily seen that (73) holds if m ∗ = maxt>TM (t − TM )e−2kb (T −TM ) is such that 8Ak p m ∗ ≥ 1. Since m ∗ = (2ekb )−1 in this case, the resulting condition reads: 4 Ak p > ekb

(74)

On Neglecting MSC Diffusion There seems to be no evidence supporting random motility (the underlying microscopical process responsible for diffusion) in MSCs or any other cell lineage involved in bone repair. Experimental studies indicate instead that most cellular events during bone healing are mainly regulated by local factors that may originate from a variety of cells. These factors produce a biochemical environment that varies at different stages of the process and determines precise cellular responses including directional cell progression, proliferation and differentiation (Cho et al. 2002). In addition to this biological argument, it is easy to see on mathematical terms that adding a diffusive term in Eq. (36) has virtually no impact on the inward motion of MSC fronts for any reasonable choice of the corresponding diffusion coefficient D. The corresponding argument can be sketched as follows: To avoid technicalities, we just show here that adding diffusion in (36) results in a small correction in the inward motion of MSC front. More precisely, let us replace (36) by ∂ ∂m + ∂t ∂x

 ∂g ∂m μm − Dm = 0; −∞ < x < ∞ , t > TM ∂x ∂x

(75)

where Dm > 0 with initial conditions (37), (38) as before. Using again the approximation ∂∂gx ≈ − 2a L x in (75), the resulting equation can be written in the form ∂m 2μax ∂m 2μa ∂ 2m − − m − Dm 2 = 0 ∂t L ∂x L ∂x

(76)

We first estimate the characteristic time τ tr needed to travel a distance L2 under transport effects. The time to fill an infinitesimal interval of the form [x − d x, x] is then given by:

123

Early Stages of Bone Fracture Healing

dx = dt . −2μax/L Integrating this expression between L/2 and a characteristic size epsilon ( ) (which can be interpreted as an intercellular space), we find that τ tr satisfies the following relationship:

τ tr

dt =

0

and we then have that: τ tr =

L 2

L dx 2μax

 L L log 2μa 2

(77)

Analogously, we estimate the characteristic time τ diff for gap filling due to diffusion as follows: L 2 diff 2 − τ ∼ (78) Dm Comparing (77) and (78) we obtain: L L

 L τ tr L Dm log 2 L Dm log 2 2Dm log = ≈ =     2 2 diff L τ 2μa L − 2μa μa L 2 2 2

(79)

L On selecting = 100 , L = 2 mm, it follows that for a typical diffusion coefficient −6 2 Dm = 10 cm /s, (79) yields:

 L 2 · 10−6 2Dm τ tr log = log(50) ≈ 6 · 10−3 = diff τ μa L 2 0.5 · 10−2 · 1.25 · 0.2 which shows the low impact of diffusion in the process under consideration. References Anitua E, Andia I, Ardanza B, Nurden P, Nurden A (2004) Antologous platelets as a source of proteins for healing and tissue regeneration. Thromb Haemost 91(1):4–15 Ayati B, Edwards C, Webb G, Wilswo J (2010) A mathematical model of bone remodelling dynamics for normal bone cell populations and myeloma bone disease. Biology Direct 5. http://www.biology-direct. com/content/5/1/28 Beenken A, Mohammadi B (2009) The fgf family: biology, pathophysiology and therapy. Nat Rev Drug Discov 8(3):235–253 Bolander M (1992) Regulation of fracture repair by growth factors. Proc Soc Exp Biol Med 200:165–170 Boyde A (2003) The real response of bone to exercise. J Anat 203(2):173–189 Brass L (2003) Thrombin and platelet activation. Chest 124(3):185–255 Butenas S, Vant Veer C, Mann K (1999) Normal thrombin generation. Blood 94(7):2169–2178 Chung R, Cool J, Scherer M, Foster B, Xian C (2006) Roles of neutrophil-mediated inflammatory response in the bony repair of injured growth plate cartilage in young rats. J Leukoc Biol 80:1272–1280 Cho TJ, Gerstenfeld LC, Einhorn TA (2002) Differential temporal expression of members of the transforming growth factor beta superfamily during murine fracture healing. J Bone Miner Res 17(3):513–520

123

L. F. Echeverri et al. Cicha I, Garlichs C, Daniel W, Goppelt-Struebe M (2004) Activated human platelets release connective tissue growth factor. Thromb Haemost 91(4):755–760 Colnot C, Huang S, Helms J (2006) Analyzing the cellular contribution of bone marrow to fracture healing using bone marrow transplantation in mice. Biochem Biophys Res Commun 350:557–561 Danforth C, Orfeo T, Mann K, BrummelZiedins K, Everse S (2009) The impact of uncertainty in a blood coagulation model. Math Med Biol 4:323–336 Dimitriou R, Tsiridis E, Giannoudis P (2005) Current concepts of molecular aspects of bone healing. Injury Int J Care 36:1392–1404 Ebenhoh O, Heinrich R (2001) Evolutionary optimization of metabolic pathways. Theoretical reconstruction of the stoichiometry of ATP and NADH producing systems. Bull Math Biol 63(1):21–55 Eghbali-Fatourechi G, Lamsam J, Fraser D, Nagel D, Riggs L, Khosla S (2005) Circulating osteoblastlineage cells in humans. N Engl J Med 352:1959–1966 Einhorn T (1998) The cell and molecular biology of fracture healing. Clin Orthop Relat Res 355(S):7–21 Fasano A, Herrero M, Lopez J, Medina E (2010) On the dynamics of the growth plate in primary ossification. J Theor Biol 265:543–553. doi:10.1016/j.jtbi.2010.05.030 Fasano A, Santos R, Sequeira A (2011) Blood coagulation: a puzzle for biologists, a maze for mathematicians. In: Modelling physiological flows, pp 44–76. Springer, Berlin Ferguson C, Miclau T, Alpern E, Helms J (1999) Does adult fracture repair recapitulate embryonic skeletal formation? Mech Dev 87:57–66 Ferguson C, Miclau T, Hu D, Alpern E, Helms J (1988) Common molecular pathways in skeletal morphogenesis and repair. Ann NY Acad Sci 857:33–42 Gaston MS, Simpson AHRW (2007) Inhibition of fracture healing. J. Bone Joint Surg 89B:1553–1560 Gerstenfeld L, Alkhiary Y, Krall E, Nicholls F, Stapleton S, Fitch J, Bauer M, Kayal R, Graves D, Jepsen K, Einhorn T (2006) Three-dimensional reconstruction of fracture callus morphogenesis. J Histochem Cytochem 54:1215–1228 Gerstenfeld L, Cullinane D, Barnes GL, Graves D, Einhorn T (2003) Fracture healing as a post-natal developmental process: molecular, spatial, and temporal aspects of its regulation. J Cell Biochem 88:873–884 Ginzburg LR, Jensen CXJ (2004) Rules of thumb for judging ecological theories. Trends Ecol Evol 19(3):121–126 Graham JM, Ayati BP, Holstein SA, Martin JA (2013) The role of osteocytes in targeted bone remodeling: a mathematical model. PLoS One 8(5):e63884. doi:10.1371/journal.pone.0063884 Groothuis A, Duda G, Wilson C, Thompson M, Hunter M, Simon P, Bail H, van Scherpenzeel K, Kasper G (2010) Mechanical stimulation of the pro-angiogenic capacity of human fracture haematoma: involvement of vegf mechano-regulation. Bone 47:438–444 Guria G, Herrero M, Zlobina K (2009) A mathematical model of blood coagulation induced by activation sources. Discret Contin Dyn Syst 25(1):175–194 Guria GT, Herrero MA, Zlobina KE (2010) Ultrasound detection of externally induced microthrombi cloud formation: a theoretical study. J Eng Math 66:293–310 Guria K, Gagarina A, Guria G (2012) Instabilities in fibrinolytic regulatory system. Theoretical analysis of blow-up phenomena. J Theor Biol 304:27–38 Guy R, Fogelson A, Keener J (2007) Fibrin gel formation in a shear flow. Math Med Biol 24:111–130 Hemker HC, Kerdelo S, Kremers RMW (2012) Is there value in kinetic modelling of thrombin generation? J Thromb Haemost 10:1470–1477 Hollinger J, Onikepe A, MacKrell J, Einhorn T, Bradica G, Lynch S, Hart C (2008) Accelerated fracture healing in the geriatric, osteoporotic rat with recombinant human platelet-derived growth factor-bb and an injectable beta-tricalcium phosphate/collagen matrix. J Orthop Res 26:83–90 Hughes S, Hicks D, Okeefe R, Hurwitz S, Crabb I, Krasinskas A, Loveys L, Puzas J, Rosier R (1995) Shared phenotypic expression of osteoblasts and chondrocytes in fracture callus. J Bone Miner Res 10:533–544 Javierre E, Vermolen F, Vuik C, Van der Zwaag S (2009) A mathematical analysis of physiological and morphological aspects of wound closure. J Math Biol 59:605–633. doi:10.1007/s00285-008-0242-7 Kameth S, Blann AD, Lip GYH (2001) Platelet activation: assessment and quantification. Eur Heart J 22:1561–1571 Kiem J, Borberg H, Iyengar GV et al (1979) Water content of normal human platelets and measurements of their concentrations og cu, fe, k and zn by neutron activation analysis. Clin Chem 21(5):705–710

123

Early Stages of Bone Fracture Healing Komarova S, Smith R, Dixon J, Sims S, Wahl L (2003) Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodelling. Bone 33:206–215. doi:10.1016/ S8756-3282(03)00157-1 Malizos N, Papatheodorou L (2005) The healing potential of the periosteum. molecular aspects. Injury 36(S3):S13–S19 Mann K, Butenas S, Brummel K (2003) The dynamics of thrombin formation. Arterioscler Thromb Vasc Biol 23:17–25 Mc Dougall S, Dallon J, Sherratt J, Maini P (2006) Fibroblast migration and collagen deposition during thermal wound healing: mathematical modelling and clinical implications. Philos Trans R Soc 364:1385–1405. doi:10.1098/rsta.2006.1773 McKibbin B (1978) The biology of fracture healing in long bones. J Bone Joint Surg B 2(60):151–162 Moore SR, Saidel GM, Knothe U, Knothe Tate ML (2014) Mechanistic, mathematical model to predict the dynamics of tissue genesis in bone defects via mechanical feedback and mediation of biochemical factors. PLoS Comput Biol 10(6):e1003604. doi:10.1371/journal.pcbi.1003604 Moreo P, Garcia Aznar J, Doblare M (2009) Bone ingrowth on the surface of endosseous implants. part i: mathematical model. J Theor Biol 260:1–12 Murphy K, Hall C, Maini P, Mc Cue S, Sean Mc Elwain D (2012) A fibrocontractive mechanochemical model of wound closure incorporating realistic growth factor kinetics. Bull Math Biol 74:1143–1170. doi:10.1007/s11538-011-9712-y Orfeo T, Butenas S, Brummel-Ziedins K, Mann K (2005) The tissue factor requirement in blood coagulation. J Biol Chem 280(52):42887–42896 Pivonka P, Dunstan C (2012) Role of mathematical modeling in bone fracture healing. BioKEy Rep 1(221):2012. doi:10.1038/bonekey.221 Prokharau P, Vermolen F, Garcia-Aznar J (2012) Model for direct bone apposition on a pre-existing surface during peri implant osteointegration. J Theor Biol 304:131–142 Rodak BF, Fritsma GA, Keohane E (2011) Hematology: clinical principles and applications, 4th edn. Saunders, USA Rozman P, Boita Z (2007) Use of platelet growth factor in treating wounds and soft-tissue injuries. Acta Dermatovenerol 16(4):156–165 Rumi M, Deol G, Singapuri K, Pellegrini VD Jr (2005) The origin of osteo- progenitor cells responsible for heterotopic ossification following hip surgery: an animal model in the rabbit. J Orthop Res 23:34–40 Sahni A, Francis CW (2000) Vascular endothelial growth factor binds to fibrinogen and fibrin and stimulates endothelial cell proliferation. Blood 96:3772–3778 Schmidt-Bleek K, Schell H, Schulz N, Hoff P, Perka C, Buttgereit F, Volk H, Lienau J, Duda G (2012) Inflammatory phase of bone healing initiates the regenerative healing cascade. Cell Tissue Res 347:567–573 Shapiro F (2008) Bone development and its relation to fracture repair. the role of mesenchymal osteoblasts and surface osteoblasts. Eur Cells Mater 15:53–76 Stefanini MO, Wu FT, Mac Gabhann F, Popel AS (2008) A compartment model of VEGF distribution in blood, healthy and diseased tissues. BMC Syst Biol. doi:10.1186/1752-0509-2-77 Stockmayer W (1943) Theory of molecular size distribution and gel formation in branched-chain polymers. J Chem Phys 11(2):45–55 Street J, Winter D, Wang J, Wakai A, McGuinness A, Redmond H (2000) Is human fracture hematoma inherently angiogenic? Clin Orthop 378:224–237 Takahara M, Naruse T, Takagi M, Orui H, Ogino T (2004) Matrix metalloproteinase-9 expression, tartrateresistant acid phosphatase activity, and dna fragmentation in vascular and cellular invasion into cartilage preceding primary endochondral ossification in long bones. J Orthop Res 22:1050–1057 Tranquillo RT, Zigmund SH, Lauffenburger DA (1988) Measurement of the chemotaxis coefficient for human neutrophils in the under-agarose migration assay. Cell Motil Cytoskelet 11:1–15 Van den Besselaaar AM (1996) Precision and accuracy of the international normalized ratio in oral anticoagulant control. Haemostasis 26(2s):248–265 Wagenwood R, Hemker PW, Hemker HC (2006) The limits of simulation of the clotting system. J Thromb Haemost 4:1331–1338 Weisel JW, Veklich Y, Gorkun O (1993) The sequence of cleavage of fibropeptides from fibrinogen is important for protofibrin formation and enhancement of lateral aggregation. J Mol Biol 232:285–297 Wiltzius P, Dietler G, Kanzig W, Haberli A, Straub P (1982) Fibrin polymerization studied by static and dynamic light scattering as a function of fibrinopeptide a release. Biopolymers 21:2205–2223

123

L. F. Echeverri et al. Yon YR, Won JE, Jeon E, et al. (2010) Fibroblast growth factors: biology, function and applications for tissue engineering. J Tissue Eng 1 Yoo J, Johnstone B (1998) The role of osteochondral progenitor cells in fracture repair. Clin Orthop Relat Res 3(55):S73–S81

123