A mathematical model of the process of ligament repair: Effect of cold therapy and mechanical stress

Journal of Theoretical Biology 302 (2012) 53–61

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A mathematical model of the process of ligament repair: Effect of cold therapy and mechanical stress Rosy Paola Ca´rdenas Sandoval a,b, Diego Alexander Garzo´n-Alvarado a,n, Ange´lica Maria Ramı´rez Martı´nez c a

´, Colombia Facultad de Ingenierı´a, Universidad Nacional de Colombia, Grupo de Modelado y Me´todos Nume´ricos en Ingenierı´a (GNUM), Cra 30 45-03, Ed 407, Of 202, Bogota ´n, Grupo de Investigacio ´n Movimiento Corporal: salud, discapacidad y educacio ´n, Colombia Facultad de Fisioterapia, Escuela Colombiana de Rehabilitacio c Facultad de Ingenierı´a, Universidad Central de Colombia, Colombia b

a r t i c l e i n f o

abstract

Article history: Received 21 October 2011 Received in revised form 18 January 2012 Accepted 24 January 2012 Available online 22 February 2012

This article proposes a mathematical model that predicts the wound healing process of the ligament after a sprain, grade II. The model describes the swelling, expression of the platelet-derived growth factor (PDGF), formation and migration of fibroblasts into the injury area and the expression of collagen fibers. Additionally, the model can predict the effect of ice treatment in reducing inflammation and the action of mechanical stress in the process of remodeling of collagen fibers. The results obtained from computer simulation show a high concordance with the clinical data previously reported by other authors. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Ligament healing Mathematical model Reaction-diffusion equations Mechanical stress Remodeling of collagen

1. Introduction The partial tear of the ligaments is a common injury affecting athletes, physically active and sedentary people. Recent data from the U.S. population, reported an injury rate of 36,9 per 100,000 people of an anterior cruciate ligament (ACL) and 9,1 for other ligaments of the knee (Davenport, 2010). This injury is called clinically as sprain grade II and is recognized by the presence of symptoms such as pain, swelling, edema, warmth, flushing, difficulty in weight-bearing, limited range of motion and joint instability (De Vita and Slaughter, 2007; Ivins, 2006). Biomechanically, there is a stress generated in the collagen fibers exceeding its elastic limit until reaching their breaking point (Wang, 2006; Subit et al., 2008). During the sprain and its recovery, there are several biological stages. First, it triggers a hemostatic phase, resulting in the formation of tissue swelling (Cotran et al., 1999). In this phase the platelets lie on the spaces of the matrix forming a first cap (Frank et al., 1999). Then it forms the expression and diffusion of chemicals substances released by the contact between platelets and the injured extracellular matrix. Its function is to attract repairing cells towards the area of tissue injury. In the case of the ligament, several growth factors are released, including and the

n

Corresponding author. Tel.: þ57 1 3165320. E-mail address: [email protected] (D.A. Garzo´n-Alvarado).

0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2012.01.035

most important, the platelet-derived growth factor (PDGF). These factors are chemotactic mediators responsible for activating the proliferation and migration of fibroblasts into the injury area, third phase of the repairing process (Cotran et al., 1999; Woo et al., 2006). Once the fibroblasts proliferate and migrate into the injury area, they initiate the formation of new components of the extracellular matrix, including immature collagen type III, glycoproteins and elastin. It should be noted, the random organization of cells in the space of the injured matrix, which generates a random distribution of new collagen fibers (Zammit, 2005; Ng, 2002; Provenzano and Vanderby, 2006). To speed up the healing process there has been several studies on the effects of physical and mechanical means in the biological tissue repairing. In the ligament, the application of cold therapy has proved benefits in reducing bleeding, swelling and pain, and improves the function, demonstrated through clinical practice (Ng, 2002; Cameron, 2003; Nobes et al., 2000; Edson, 2003; Frommer, 2009). Its action is due to the production of constricting blood vessels, increasing blood flow and contributing to the removal of necrotic tissue and fluids accumulated (Cameron, 2003; Nobes et al., 2000). Likewise, it has been reported the favorable effect of the mechanical means in the stages of ligament repair through in vitro studies. For example, the application of therapeutic ultrasound has shown an increase in the proliferation of fibroblasts and consequently an increase in the synthesis of proteins, mainly collagen (Doan, 1999). In addition, the use of

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mechanical stress, promotes the alignment of the new collagen fibers (Baaijens et al., 2009; Woo et al., 2000). These processes improve the biological and mechanical properties of the ligament (Kjaer and Magnusson, 2008; Park, 2006). In vitro experiments have allowed the understanding of the stages of ligament repair process from the histological and biochemical point of view (Woo et al., 2006; Enoch and Leaper, 2007; Frank et al., 1999). To complement the experimental data, computational studies have been developed to describe the biomechanical behavior of the tissue. Under the computational work it is possible to determine the stresses and strains that a tissue has to different types of loads, using constitutive relations and geometric shapes close to reality in vivo (Pioletti et al., 1998; ˜ a et al., 2006). Also, computer models have Weiss et al., 2005; Pen ˜a been used to quantify the mechanical properties in grafts (Pen et al., 2005) and compare their performance with healthy tissue (Abramowitch et al., 2004). Given the high costs of experimentation in vitro, the difficulty to access a study directly on the population and the ethical factors involved in human experimentation, the construction of computer models to reproduce complex biological phenomena has been an alternative. From this perspective, the objective of this study is to formulate a mathematical model to predict the ligament repair process and the effect of treatment with ice and mechanical stresses. In this regard we have studied the effect of cold on the swelling, the synthesis of chemotactic factors, proliferation and migration of fibroblasts and collagen production. In addition, we studied the effect of mechanical stress on the remodeling process of the fibers. This model was implemented computationally using the finite element method to simulate the process of repairing a sprain grade II. The results show a high concordance with clinical experience and the values reported by other researchers.

2. Materials and methods Biological events considered in the ligament repair, after suffering a sprain grade II are: swelling, release of the plateletderived growth factor (PDGF), fibroblast migration, formation and remodeling of the collagen (Cotran et al., 1999; Enoch and Leaper, 2007; Frank et al., 1999; Cumming et al., 2010; Provenzano, 2007). The approximate time of each of these stages are shown in Fig. 1 The first stage, hemostasis, includes the immediate activation of the platelets to form the clot that stops the fluid extravasation caused by the rupture of blood vessels and lymphatics (Frank et al., 1999). Thus, during the following 12–24 h after the injury (Ma´rquez Arabia and Ma´rquez, 2009), swelling is due to congestion and edema, this is, the local increase in blood volume, subtle growth of the cells and separation of the elements of the extracellular matrix (Cotran et al., 1999; Frank et al., 1999). The collection of blood at the area of the sprain makes platelets interact with the injured extracellular matrix to synthesize the platelet-derived growth factor (PDGF), a chemotactic substance essential for the activation of the proliferation and migration of fibroblasts to the area of injury (Enoch and Leaper, 2007; StreckerMcGraw et al., 2007). In approximately two to four days (Enoch and Leaper, 2007), fibroblasts travel from the epiligament, surface layer that surrounds the fiber bundles, to the injury area (StreckerMcGraw et al., 2007; Woo et al., 1998). These cells slip through the fibrin strands and the collagen fibers to reach the wound, once there, they produce fibronectin, hyaluronan and later, collagen and proteoglycans (Enoch and Leaper, 2007). Then, the collagen is synthesized by fibroblasts (Enoch and Leaper, 2007). This protein is the most important in tissue repair because it gives rigidity. This stage seeks to connect collagen

Fig. 1. Stages of the repairing process followed by a partial rupture of the ligament fibers. Adapted from (Ng, 2002; Enoch and Leaper, 2007).

fibrils residual between end to end by the synthesis of immature collagen type III (Provenzano and Vanderby, 2006) following a random and disorganized pattern (Woo et al., 2000). In the sprain grade III it reaches its maximum in the third week (Ng, 2002) and in sprain grade II it approximates to 9 days (Provenzano, 2005). Once the scar has matured, product of the remodeling of the components of the extracellular matrix during months and years, the collagen bundles increase its diameter. However, these collagen fibers will never achieve their original strength of the normal tissue (Woo et al., 2006; Enoch and Leaper, 2007; Frank et al., 1999). Frank et al., (1999) found only 30–40% of the mechanical properties of the normal ligament to ligament healed; the potential cause is related to poor alignment of collagen fibers. The remodeling process and the effect of mechanical load is a constant topic in current research (Baaijens et al., 2009). For this reason, is precise, to research about the mechanisms to improve the healing process and alignment of the fibers. In the next section, we present the mathematical model about main phases of ligament healing, when it has suffered partial rupture of his fibers. 2.1. Mathematical model We present the mathematical model that describes the repair phase of an extra-articular ligament, as the medial collateral ligament of the knee, nine days after the partial rupture of the fibers. In the model we include changes by the application of physical and mechanical means such as cold and mechanical stress. 2.1.1. Swelling The first phenomenon that occurs after the rupture of the fibers is the production of hemorrhage followed by congestion

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and edema in the affected area, known as swelling (Frank et al., 1999). For this purpose we use the variable C, which corresponds to the concentration of the fluids released by the blood and lymphatic system. Therefore, it is assumed that the concentration C evolves according to the expression given by Eq. (1): DCðx,tÞ þCðx,tÞdivðvÞ ¼ f ðx,tÞ Dt

ð1Þ

where DCðx,tÞ=Dt is the material derivative of the concentration of C(x,t) (Bonet and Wood, 2008). This article assumes that the swelling occurs when the concentration of the fluids in the area of injury reaches a saturation level Csaturation. Otherwise, when C oCsaturation the volume will remain constant, this is div(v) ¼0. A few minutes after the partial ligament rupture (Ma´rquez Arabia and Ma´rquez, 2009), the area is congested by fluids that reach the saturation level Csaturation, and therefore, the only way to increase the amount of fluid in the edema is by the local increase of the volume of the wound, keeping the concentration of liquid at a constant level, this is DCðx,tÞ=Dt ¼ 0. Then, there is an isotropic growth of the volume due to the increased presence of fluid in the wound when C¼ Csaturation. Therefore, we propose that the strain rate (Mase and Mase, 1999) corresponds to: dswelling ðx,tÞ ¼ trðdswelling ÞI

ð2Þ

where, trðdswelling ÞI ¼ divðvÞ ¼

f ðx,tÞ ¼ gðx,tÞ C

where the function (x,t) is the given by: 8 fðT 0 ðx,tÞÞnhswelling ðtÞ > > > > < fðT 0 ðx,tÞÞnhconstant ðtÞ gðx,tÞ ¼ > > fðT 0 ðx,tÞÞnhShrink ðtÞ > > : 0

ð3Þ

function of local growth, which is if t 1 r t rt 2 if t 2 r t rt 3 if t 3 r t rt 4

ð4Þ

Otherwise

where hi(t) are dimensionless functions of time that quantify the swelling. The swelling, first function appears when the time is between t1 and t2. The swelling constant, second function appears when the time is between t2 and t3. The shrink, third function appears when the time is between t3 and t4. Finally, the function is zero to other case. For its part, j(T(x,t)) is a function, depending on the temperature, that quantifies the concentration of fluid present in the swelling, which is given by:

jðTðx,tÞÞ ¼ j0

Tðx,tÞT 0 T 1 T 0

ð5Þ

where j0 is a constant that quantifies the rate of swelling, T(x,t) is the temperature at each point of the sprain, T0 is the optimum temperature at which the process of ligament repair should be carried out and TN is the body temperature, at each instant of time. For its part, in the case of swelling in the time interval [t1, t2], the wound is congested and the fluid increases in quantity (Strecker-McGraw et al., 2007), so we assume: hswelling ðtÞ ¼ a1 

t

b1

ð6Þ

In the case where the maximum congestion and volume is reached, it starts the vasoconstriction and coagulation of the plasma present in the congestion (Cotran et al., 1999). Therefore, there is a time interval [t2, t3], in which the swelling ceases and is assumed to remain with a constant volume given by: hconstant ðtÞ ¼ 0

ð7Þ

Then, the liquid, the waste and clotted blood leave the wound, in the time interval [t3, t4], so there is a reduction in the swelling

55

(Enoch and Leaper, 2007), expressed as: hshrink ðtÞ ¼

t

b2

a2

ð8Þ

where each of the ai and bi are constant values. 2.1.2. Release of the platelet-derived growth factor (PDGF) Once there is contact between the platelets and the damaged extracellular matrix, the platelets release, mainly, the growth factor derived from the platelets PDGF (Rozman and Bolta, 2007). This factor travels from the site of injury to the epiligamet with the aim of activating the migration and proliferation of fibroblasts (Cotran et al., 1999; Benani, 2008). The evolution of the growth factor is represented as: @G 2 þ r UðvGÞ ¼ DG r G þbðC p ,x,tÞdðG,x,tÞ @t

ð9Þ

where G is the concentration of the platelet-derived growth factor (PDGF), DGr2G corresponds to the diffusion term which follows Fick’s law, b(Cp,x,t) indicates the release of the factor PDGF by the platelets factor (where Cp is the concentration of platelets) and d(G,x,t) indicates the disappearance of the growth factor due to the average lifetime before degradation. On the other hand, DG is the diffusion coefficient and v is the local velocity of each of the points in the domain that depends on the process of swelling. Thus, the release term is approached to: bðC p ,x,tÞ ¼ gC p eat

ð10Þ

where g is a constant that quantifies the rate of release of the growth factor by the platelets and the term e  at quantifies the action of the platelets that decreases its effectiveness of release with its time of life (Thomopoulos et al., 2010). In this case the concentration of the platelets Cp is considered constant, because the blood and the clot in the wound do not change its concentration over time (Thomopoulos et al., 2010). On the other hand, the term of degradation is given by (11): dðG,x,tÞ ¼

lnð2Þ

tG

G

ð11Þ

where tG is the average life time of the growth factor (Gay and Winkles, 1991; Brouwers et al., 2006). The equation of growth factor concentration is completed by the boundary condition that considers the release of PDGF when it comes in contact with the platelets in the injured area of collagen, this is: ðrGÞUn ¼ hðtÞ

ð12Þ

where (rG) is the flow of PDGF released by platelets at the edges of the lesion, n is the normal at the contour directed out of the area of injury, (see Fig. 3), and h(t) is the release function of PDGF in the contour injured by platelets and is given by: hðtÞ ¼ j

xi i

x þt n

ð13Þ

where j quantifies the release gradient of the growth factor (Garzo´n-Alvarado et al., 2009) and xi quantifies the maximum time of platelets release in the boundary, t represents the repair time and n is a constant that quantifies the slope of the function. 2.1.3. Migration of fibroblasts When the growth factor diffuses and reaches the area of epiligament, in this area the fibroblasts are released by the action of PDGF (Enoch and Leaper, 2007; Lo et al., 2002). Therefore, the release of fibroblast is given by the boundary condition (see Fig. 3): ðrFÞUn ¼ rðG,tÞ

ð14Þ

´rdenas Sandoval et al. / Journal of Theoretical Biology 302 (2012) 53–61 R.P. Ca

56

where F represents the concentration of fibroblasts, n is the normal at the contour directed out of the area of epiligament, (see Fig. 3), and r is a function that depends on the amount of growth factor present on the boundary and that induces the release of fibroblasts (Hannafin et al., 1999), we can see that the formation of fibroblasts depends, only, on the amount of PDGF that is in the epiligament, this is: rðG,tÞ ¼ z  Gðx,tÞ

ð15Þ

where, z is a constant that quantifies the flow of fibroblasts. Once the fibroblasts are released, it is assumed that the cell concentration, at each time step, can be described by the equation: @F lnð2Þ þ r UðvFÞ ¼ rðDF rFF wðGÞrGÞ F @t tF

ð16Þ

where DF is the diffusion coefficient of fibroblasts, w is sensitivity to the chemotaxis and tF is the average lifespan of a fibroblast (Brouwers et al., 2006). This equation takes into account the chemotaxis of fibroblasts by the action of the PDGF factor. Moreover it also takes into account the diffusive flow and degradation and apoptosis of fibroblasts. The variable w corresponds to the chemotaxis sensitivity which is defined as (Hannafin et al., 1999):

w¼

an

2.1.5. Remodeling of collagen fibers By applying mechanical loads of tension-type, cells and collagen fibers are oriented in the direction of the stresses to allow the homogeneous formation and organized in the new collagen fibers (Woo et al., 2000). This process improves the mechanical and biological properties of the ligament (Kjaer and Magnusson, 2008; Park, 2006). In this model the orientation of the fibers is a function of the mechanical stress imposed, namely:  f ðr,nÞðoo0 Þ if absðoo0 Þ 4 0 do ¼ ð21Þ dt 0 Other case where o0 is the optimum angle of the fiber, f(r,n) is a function that depends on mechanical stimulation, and it is given by Carter et al. (1988): !1=m N X m f ðr,nÞ ¼ b ni s ð22Þ i¼1

where b and m are constants of calibration of the model, N is the number of load cases, ni is the average number of cycles per time pffiffiffiffiffiffiffiffiffi unit and per case load, s ¼ 2 2EU is the equivalent stress in each fiber. In addition, E is the modulus of elasticity, U is the strain energy per unit volume. Then, the remodeling collagen fibers occur one year after injury approximately (Frank et al., 1999), in the model imply a time step Dt ¼585 min.

ð17Þ

ðbn þGÞ2

2.2. Numerical implementation

where an and bn are constants of the chemotactic term (Javierre et al., 2009). 2.1.4. Immature collagen formation Once the fibroblasts have migrated into the injured area it begins the process of collagen formation (Provenzano, 2005). Fibroblasts deposit bundles of collagen type III from undamaged collagen fibers to join the ends with the intact contours. This deposition is initially done randomly (Provenzano and Vanderby, 2006; Woo et al., 2000). This model assumes that the deposition of collagen depends on the number of fibroblasts present at each point of the domain. The formation of each fiber is assumed to be a vector that to have magnitude given by: rðt þ dtÞ 8 T > < rðtÞ þ ðr col ðh,tÞ  cosðoÞ,r col ðh,tÞ  sinðoÞÞ ¼ rðtÞ > : 0

if 9r col ðh,tÞ9 o 1 and F 4 F threshold, if 9r col ðh,tÞ9 Z 1, Otherwise,

ð18Þ

The implementation was developed on a personal computer with Intel Core 2 Duo of 2.6 Hz, 2 GB in RAM, using the finite element method (FEM) trough software Fortran. The geometry of the area of injury was taken from the Medial Collateral Ligament (MCL) (Woo et al., 1998). The domain has dimensions of 20 mm long and 9.3 mm in wide, these data are close to 30% of fibers injured and represents the main area of injury, middle portion of the tissue (Doschak and Zernicke, 2005; Laws, 1988) (Fig. 2). It consists of a two-dimensional mesh of fournode bilinear elements. In addition the domain is divided into two surfaces representing the area of residual collagen fibers and the intact (area I) and the area of injury (area II) (Fig. 3). In the intact area there are 18,360 elements and in injury domain we have 1712. The total number of elements in the entire domain is 20,072. As for the initial conditions, they are assumed to be null for all variables. The boundary conditions are observed in the Fig. 3. The calculation of the swelling is done both in the area I and in

where rcol(h,t) is the magnitude of the vector representing the collagen fiber, o represents the random angle of orientation of the fiber and Fthreshold is the threshold value of the concentration of fibroblasts which start the process of formation of immature collagen. In the expression (18) rcol is given by: r col ðh,tÞ ¼

vgrowth  dt hðx,tÞ

ð19Þ

where vgrowth is the growth rate of the collagen fiber thanks to the deposition of fibroblasts, dt is the time difference and in the case of finite elements, h(x,t) is the value of the equivalent diameter of the element in each time step and at every point in space. For its part, the angle that takes the fiber is given by:

o ¼ p  randomð½0,1Þ

p 2

ð20Þ

where random is a function that returns a pseudo-random value uniformly distributed on the interval [0,1]. Therefore, the fiber   can be oriented at an angle between  p2 , p2 . It should be noted that once a collagen fiber takes a specific direction, following the equation, this orientation is constant throughout the simulation.

Fig. 2. The mesh of medial collateral ligament (MCL). (a) View of MCL, adapted from (Sobotta, 1999). (b) Injured area of the MCL in the middle portion (red triangle) shows the congestion of blood. (c) It shows a zoom of the axi-symmetric area of injury. (d) two-dimensional axi-symmetric mesh with dimensions of 20 mm  9.3 mm, the color gray surface to a height of 4.75 mm represents the undamaged fibers so on, represents the terminal fibers after injury. The red area symbolizes the injury area. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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area II in order to maintain compatibility of mechanical strain in the two domains mentioned. For its part, the calculation of the differential equations for the growth factor, fibroblasts and collagen fibers was conducted in Area II, which corresponds to the injured area. The simulation time represents the following nine days to the partial rupture of the ligament fibers, including the times of the phenomena: decreased swelling, increased in the release rate of the growth factor PDGF, migration of fibroblasts, production and collagen remodeling. A mechanical process is assumed where there is a turnover of collagen fibers as a function of mechanical

57

stress. We used a time step Dt¼25 min, during the first nine days, and Dt ¼585 min, for the following year of fiber remodeling. For its part, to calculate the temperature of the ligament, we carried out a simplification, therefore, we assume a constant temperature at the top and bottom and a null flow in the right and left side as shown in Fig. 4. 2.2.1. Parameters From the information available in literature (Brouwers et al., 2006; Javierre et al., 2009; Creaney and Hamilton, 2008) and by numerical analysis the parameters used in the model are summarized in Table 1.

3. Results

Fig. 3. Boundary conditions. Area I represents the intact fibers and terminals. Area II corresponds to the area of injury. The top line can be seen as the surface layer of the epiligament.

The first stage of healing and recovery from a sprain is swelling. This fulfills an important function: platelet deposition in the area of injury through the process of pre-congestion (Cotran et al., 1999). In the model, swelling reaches its maximum level in the first day after the injury. In Fig. 5, row (a) shows a lesion that is not being treated with any therapy and therefore inflammation increases more than in Fig. 5(b) and (c), which use external cooling means to control the swelling. In these cases it is considered that the surface temperature is, hypothetically, zero degrees Celsius, and the internal temperature (in the central part of the ligament) is 25–35 1C for row 5(b) and 5(c), respectively. In these graphs, the effect of temperature decreases swelling by the third day (Cameron, 2003; Nobes et al., 2000) until it disappears

Fig. 4. Boundary conditions. Area I represents the intact fibers and terminals. Area II corresponds to the area of injury. The top line can be seen as the surface layer of the epiligament.

Table 1 Parameters used in the model. Parameters

Value

Unit

Reference

j a1

8.0  10  4 1.0 1400.0 0.28 35000.0 3.5  10  3 1.44  10  8 178  103 3.4  10  5 126 0.0001 7200.0 0.0005 1.39  10  3 2  10  3 2.78  10  5 2  10  3 1.44  104 13.0 3.97  10  4 0 0

min  1 Dimensionless min Dimensionless min mm2/min ng/plateletsnmin platelets/mm3 min  1 min ng/mm4 min fibroblasts/mmnng mm2/min ng/mm3 ng/mm  day ng/mm3 min fibroblasts/mm3 mm/min 1C rad

Numerical Experimentation Numerical Experimentation (Enoch and Leaper 2007) Numerical Experimentation Numerical Experimentation (Brouwers et al., 2006; Javierre et al., 2009) (Garzo´n-Alvarado et al., 2009) (Gay and Winkles 1991) (Lo et al., 2002) (Gay and Winkles 1991) (Garzo´n-Alvarado et al., 2009) (Thomopoulos et al., 2010) Numerical Experimentation (Javierre et al., 2009) (Javierre et al., 2009) (Javierre et al., 2009) (Javierre et al., 2009) (Clark and Singer, 2000) Numerical Experimentation Numerical Experimentation Numerical Experimentation Numerical Experimentation

b1

a2 b2 DG

g Cp a ¼ Ln(2)/tPlaquetas

tG f x z DF

w an bn

tF Fthrshold vcrec T0

o0

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Fig. 5. Concentrations of the platelet derived growth factor PDGF over time, simultaneously we evidence swelling of the tissue. Row (a) sprain not treated with temperature. Row (b) Used the following values of Fig. 4(a) TN ¼25 1C, Tsurface ¼0 1C. Row (c) used TN ¼ 35 1C, Tsurface ¼ 0 1C. Each column represents time in days.

Fig. 6. Concentrations of fibroblasts over time. Row (a) sprain not treated with temperature. Row (b) used the following values of Fig. 4(a) TN ¼25 1C, Tsurface ¼0 1C. Row (c) used TN ¼35 1C, Tsurface ¼0 1C. Each column represents time in days.

in the fifth day. In addition, this figure shows that platelets can release the growth factor PDGF required to activate the migration of fibroblasts. Fig. 5(a) shows a high concentration near the area of rupture of the collagen fibers. Between the first and third day (columns t ¼1.7 and 3.5 days) the concentration of the growth factor PDGF rises and remains at a range of 0.5–1.0 ng/mm3. Subsequently, from the fifth day onwards the PDGF values decrease in time. The function of the growth factor PDGF is to activate the migration of fibroblasts from the epiligament (see Fig. 4) to direct the cells (fibroblasts) to the area of injury. When the values of concentration of the growth factor PDGF are maximum (see Fig. 5), the migration of fibroblasts into the injury area (see Fig. 6) becomes evident. This occurs finishing the first and third day after the injury, the values of the concentration of fibroblasts oscillate in a range of 6.00 cell/mm3 and 2.00 cell/mm3. Note that the concentration of cells is greater when using cold therapy reduces the swelling. Thus, in rows (b) and (c) in Fig. 6, there is greater concentration and greater coverage of cells at the area of the sprain, as seen at t ¼1.7, 3.5 and 5.2 days.

The maximum value of fibroblasts occurs in the third day, and approximates to 38.00 cell/mm3. In this day, the first outbreak of immature collagen fibers is observed (see Fig. 7). These cells spread in the direction of the gradient of PDGF. The area of outbreak of the collagen fiber is the vertex of union between damaged fiber line and epiligament, from this area it starts the growth of each collagen fiber to reach the vertical line of symmetry. Around the sixth day a big part of the fiber network has been formed, following a random distribution and disorganized but continuous. On the ninth day, the first immature collagen mesh has been formed. Note again that, due to ice therapy, the initial formation of collagen fibers occurs in 1.7 days (rows (b) and (c)), and the formation is completed before (t ¼5.2 days for the rows (b) and (c)) compared to the untreated sprain (row (a)). 3.1. The remodeling process Once the formation process of collagen fibers is completed, it can carry out the remodeling process to improve the mechanical

´rdenas Sandoval et al. / Journal of Theoretical Biology 302 (2012) 53–61 R.P. Ca

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Fig. 7. Immature collagen formation. Row (a) sprain not treated with temperature. Row (b) used the following values of Fig. 4(a) TN ¼ 25 1C, Tsurface ¼0 1C. Row (c) used TN ¼ 35 1C, Tsurface ¼ 0 1C. Each column represents time in days.

Fig. 8. Remodeling of the collagen fibers. Row (a) Sprain not treated mechanically, row (b) f ðr,nÞ ¼ 4:2  106 min1 and row (c).f ðr,nÞ ¼ 4:2  107 min1 .

properties of the tissue (Woo et al., 2000). The remodeling time is approximately one year (Baaijens et al., 2009; Woo et al., 2000). For this simulation, in Fig. 8, we used the following values for the remodeling: in the row (a) f(r,n)¼0, row (b)) f ðr,nÞ ¼ 4:2  106 min1 and row (c) f ðr,nÞ ¼ 4:2  107 min1 . In row (a) no mechanical loads have been used for the correction of fibers, therefore, the orientation of these is disordered conferring low resistance to the ligament. In row (b), there is a correction of the orientation of the fibers without reaching the ideal form of it. Finally, in row (c), we obtain, hypothetically, the total correction of the collagen fibers.

4. Discussion In this paper we propose a computational model that predicts the evolution of the injured ligament repair and the effect of the application of physical and mechanical means. We propose a complete model of healing that goes from the swelling, to the remodeling of the collagen fibers. In particular, the model proposes a hypothesis about the beneficial effect of ice application and mechanical stress in reducing swelling, increase of the concentration of the growth factor PDGF and fibroblast, collagen production and in the orientation of fibers as a function of mechanical stress during the remodeling process.

Here, we have studied the beneficial effect of the use of cold therapy during the stage of swelling. It has been clinically observed that the use of cold reduces the size of the inflammation by vasoconstriction. This effect is similar, from the geometrical point of view, to what happens in the contraction of engineering materials subjected to temperature decrease. For this reason we have mathematically modeled as a linear effect of temperature dependent. Under this perspective, the use of cold therapy reduces the size of the swelling (Nobes et al., 2000; Edson, 2003; Frommer, 2009). Therefore, the concentration of fibroblasts and PDGF is not reduced by dilution due to excessive growth in the area of the sprain. This important effect is definitive in the first hours and days after injury, because it allows a repair process with greater concentration of cells to form collagen earlier, compared with sprains without external treatment. Given the simplification of the biological phenomena that occur in the repairing of the ligament, due to its complexity, a number of limitations are present. First, we used data reported in the scientific literature to approximate the model to the reality of the events related to repairing the ligament. Those parameters, in which it was not possible to find its value, were estimated by numerical experimentation of the model presented here. Second, it is assumed that the migration of fibroblasts depends only on the concentration of platelet derived growth factor (PDGF). According to experimental findings this growth factor

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has a greater effect on the formation of fibroblasts compared with other growth factors such as epidermal factor (EGF) and transforming growth factor (TGF) (Woo et al., 1998). In this sense, literature reports values close to those found in this work in terms of concentrations and the rate of release of the growth factor PDGF (Thomopoulos et al., 2010). Third, we began with the fact of the pre-existence of a provisional fibrin mesh created by the kinetic reaction between thrombin and fibrinogen produced by platelets. On this mesh the fibroblasts migrate from the layer of the epiligament into the injured matrix (Flynn and McCormack, 2008). Furthermore, it should be noted that the model involves the production of collagen in terms of the number of fibroblasts deposited at each point of the domain and the random distribution of the fibers   depends on a ramdom angle between  p2 , p2 (Eq. (20)). Once the collagen fiber takes a specific direction following the equation, this orientation is constant throughout the simulation, during the first nine days. This assumption must be revised in light of experimentation. Additionally, the model allowed us to obtain the evolution of remodeling under the effect of mechanical stress. Accurate knowledge about the role that mechanical stress has on the fiber orientation of collagen during the remodeling process remains a subject of research at a computational and experimental level. In this article, we have done a ’’phenomenological’’ and linear assumption, on the action of the stress on their orientation. It is important to note that this approach does not allow estimating the stress and number of cycles at which the fibers are damaged, which should be the subject in research at a clinical level. It should also be clear that some steps were not considered in the model. One is the production of other extracellular matrix components such as proteoglycans and elastin, because tissue integrity depends mainly on the new collagen fibers (Woo et al., 2006). This process involves extracellular matrix turnover for months and years (Struijs and Kerkhoffs, 2005; Cowin, 2004), where the immature collagen (type III) is replaced by mature collagen (type I) (Enoch and Leaper, 2007). Experiments have shown that the application of mechanical stimuli on the stage of migration of fibroblasts consistently reorients the new collagen fibers (Park, 2006). On the other hand, the results correspond with clinical observations in terms of time needed to reach the stages of ligament repair. The time of tissue swelling approaches to the reported by experimental literature, this is, it begins in the first 24 h and disappears by the fifth day on average (Enoch and Leaper, 2007; Struijs and Kerkhoffs, 2005). The release of growth factor PDGF occurs in the first four hours and continues on the ninth day but with a lower concentration (Cowin, 2004). The migration of fibroblasts occurs between the first and third day (Cotran et al., 1999). Finally, the formation of immature collagen fibers occurs in nine days. According to Provenzano et al., this time is sufficient for the production of collagen type III in the sprain grade II (Provenzano, 2005). Therefore, the model’s predictions on ligament repair times are within the expected biological range. As a synthesis, the model reproduces in a simplified way the biological phenomena of ligament repair after partial rupture of the fibers. In particular it has determined the influence of physical and mechanical means in reducing swelling, the increase of concentrations of PDGF and fibroblasts, increase in the velocity of its production and of collagen and the redistribution of the fibers in a homogeneous way as a function of the stress during the remodeling stage. Therefore, the model allows us to conclude that the application of ice and mechanical stress is beneficial to the healing process, increasing the velocity of the healing and the formation of oriented collagen fibers. Therefore, this work allows

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