A Fibrocontractive Mechanochemical Model of Dermal Wound Closure Incorporating Realistic Growth Factor Kinetics

Abstract

Fibroblasts and their activated phenotype, myofibroblasts, are the primary cell types involved in the contraction associated with dermal wound healing. Recent experimental evidence indicates that the transformation from fibroblasts to myofibroblasts involves two distinct processes: The cells are stimulated to change phenotype by the combined actions of transforming growth factor β (TGFβ) and mechanical tension. This observation indicates a need for a detailed exploration of the effect of the strong interactions between the mechanical changes and growth factors in dermal wound healing. We review the experimental findings in detail and develop a model of dermal wound healing that incorporates these phenomena. Our model includes the interactions between TGFβ and collagenase, providing a more biologically realistic form for the growth factor kinetics than those included in previous mechanochemical descriptions. A comparison is made between the model predictions and experimental data on human dermal wound healing and all the essential features are well matched.

Appendices

Appendix A: Non-dimensional Equations

Applying the following non-dimensionalization,

and dropping bars, we obtain the following non-dimensional equations:

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

Appendix B: Initial Conditions

The following represent the scaled initial conditions employed in this model:

(29)

(30)

(31)

(32)

(33)

(34)

where ϵ _n =0.1, ϵ _β =ϵ _ρ =ϵ _z =0.4, controlling the steepness across the boundary, ρ _in=0.1, the initial scaled collagen density within the wound space, P _ss is the steady-state value for PDGF in the presence of fibroblasts, given by P _ss=a _P /(δ _P +δ _Pn n), P _in=a _P /δ _P , is the steady-state value for PDGF in the absence of fibroblasts, and L=1, the scaled initial position of the wound boundary.

Appendix C: Parameter Estimation

First, we estimate values for the scalings used to non-dimensionalize the variables:

L::

A typical length scale for acute dermal wounds is 1 cm.

T::

A typical length scale for time is days. Hence, T=1 day.

r::

In Murphy et al. (2011) we estimate fibroblast proliferation to be r=0.832/day.

$\theta_{nn}^{-1}$::

The carrying capacity of fibroblasts is known to be approximately 10⁶ cells/mL (Vande Berg et al. 1989). Hence, we take $\theta_{nn}^{-1}=10^{6}\ \mathrm{cells}/\mathrm{mL}$.

k/δ _ρ ::

It is known that 30% of newly synthesized collagen is degraded (Aumailley et al. 1982). Hence, δ _ρ =0.3k, such that k/δ _ρ =3.33. Bahar et al. (2004) estimates a collagen production rate of 1.75 pg/cell day.

β ₀::

Yang et al. (1999) found the initial concentration of TGFβ in the wound to be 275 ng/mL. Hence, we take β ₀=275 ng/mL.

P ₀::

Olsen et al. (1995) states that PDGF is stored in platelets at concentrations of approximately 15–50 ng/mL. Olsen et al. (1995), Haugh (2006) and Schugart et al. (2008) all propose an initial PDGF concentration of P ₀=10 ng/mL, which we adopt.

We can now apply the following non-dimensionalization:

The values for the remaining dimensional parameters are as follows. app D _n : Experiments by Sillman et al. (2003) found that fibroblasts derived from normal human dermal wounds migrate at an average velocity of 0.23–0.36 μm/min. This gives a range for the minimum wavespeed of 0.00033<D _n <0.001 cm²/day. We choose the upper limit of D _n =0.001 cm²/day.

χ::

Olsen et al. (1995) recognized that the chemotactic coefficient should predominate over the random diffusive flux. In the absence of quantitative studies, Haugh (2006) and Monine and Haugh (2008) propose that the chemotactic coefficient is three times the magnitude of the diffusivity. We chose a value for D _n =0.001 cm²/day, and controlling for the PDGF density (P ₀=10 ng/mL), this gives a chemotaxis coefficient of χ=0.03 ng/cm day.

a _χ ::

Olsen et al. (1995) notes that experimental data suggests that the half-maximal response of fibroblasts to PDGF-mediated chemotaxis occurs at a concentration of 2 ng/mL. Thus, we take a _χ =2 ng/mL.

a _nβ ::

Strutz et al. (2001) found TGFβ to increase fibroblast proliferation by 2–3 times. Hence, we assume that a _nβ =2/β ₀.

α::

Desmouliere et al. (1993) found that culturing fibroblasts in the presence of TGFβ increased the percentage of cells expressing α-SMA from 7.5% to 45.3%, representing an activation of 37.8% of fibroblasts, and is consistent with other estimates (Masur et al. 1996; Moulin et al. 1996). This experiment occurred over a one week period, with a TGFβ dose of 5–10 ng/mL. This gives a range for the activation of 0.0054<α<0.0108/day (ng/mL). We choose the upper limit of α=0.0108/day (ng/mL).

a _mσ ::

The myofibroblast growth rate is lower than that of normal dermal fibroblasts, with myofibroblast growth approximately 50% that of fibroblasts (Vande Berg et al. 1989). Thus, we take the myofibroblast proliferation to be half that of fibroblasts, such that a _mσ =0.5r.

a _mβ ::

We assume that myofibroblasts experience the same increase in proliferation due to TGFβ as fibroblasts. Hence, a _mβ =a _nβ .

θ _m ::

The doubling time of fibroblasts is approximately 18 hours (Olsen et al. 1995). We assume that the doubling time of myofibroblasts is the same as that for fibroblasts. Hence, this gives a natural cell death rate for the myofibroblasts of θ _m ≈0.90.

θ _mm ::

As myofibroblasts are roughly twice the size of fibroblasts (Masur et al. 1996), we assume that myofibroblasts have half the carrying capacity of fibroblasts, i.e., θ _mm =2θ _nn =(0.5×10⁶)⁻¹.

D _β ::

Using known estimates of the molecular weight of epidermal growth factor (EGF) and TGFβ (Cell Signaling Technology 2010) and the diffusivity of epidermal growth factor (Thorne et al. 2004), we were able to determine the diffusivity of TGFβ using the Stokes–Einstein formula, such that D _β ≈0.0254 cm²/day.

a _β ::

Experiments by Wang et al. (2000) give the range for TGFβ production by fibroblasts as 0.125<a _β <0.525×10⁻⁶ ng/(cell day). We choose the lower limit, such that a _β =0.125×10⁻⁶ ng/(cell day).

η, π, ζ::

On a percentage basis, myofibroblasts produce roughly twice the collagen that is synthesized by fibroblasts (Kim and Friedman 2009; Moulin et al. 1998; Olsen et al. 1995). Hence, we choose η=2. There is a similar trend for myofibroblast synthesis of TGFβ (see Kim and Friedman, 2009) and based on these relations, we assume the same is true for myofibroblast production of collagenase. Hence, π=ζ=η=2.

b _β ::

Using estimates from Dale et al. (1995), inhibition of TGFβ synthesis is assumed to be b _β =5/β ₀.

a _βz ::

Using order of magnitude approximation, we estimate the activation of TGFβ by collagenase to be ∼O(0.1) when non-dimensionalized. Thus, a _βz =0.0014 mL/ng day.

a _βm ::

We assume that the amount of TGFβ activated from matrix stores is of the same order of magnitude as the amount of TGFβ activated by collagenase following non-dimensionalization, i.e., O(0.1). Hence, we estimate the activation of TGFβ by myofibroblasts to be 4.37×10⁻⁹ mL day/cell.

δ _β ::

The TGFβ decay rate was estimated from the exponential phase of the data from Yang et al. (1999), giving a rate of δ _β ≈0.354/day.

D _P ::

Haugh (2006) states that the diffusion coefficient for PDGF in aqueous solution is estimated at 10⁻⁶ cm²/s (0.0864 cm²/day), or twice the value taken by Olsen et al. (1995). However, Haugh (2006) then states that diffusion of cytokines in tissue is much slower than in solutions, and that the diffusion of PDGF in the dermis is approximately one thirtieth of its value in solution. Thus, the diffusion coefficient for PDGF is taken to be D _P =0.00288 cm²/day.

δ _P ::

Olsen et al. (1995), Haugh (2006) and Monine and Haugh (2008) all consider PDGF decay to be O(1)/day. We use the value given by Haugh (2006) and Monine and Haugh (2008) of δ _P =2.4/day.

a _P ::

The range suggested by Olsen et al. (1995) for the production of PDGF (depending upon the cellular density, which ranges from 10⁴–10⁶) is 4–400 ng/cm³ day, while Haugh (2006) proposes limits of 4.8–48 ng/cm³/day, which we see encapsulates the lower end of the parameter range suggested by Olsen et al. (1995). Both Haugh (2006) and Monine and Haugh (2008) use the value of a _P =24 ng/cm³/day so that the production rate of PDGF balances the degradation rate in the absence of fibroblasts (where a _P =δ _P P ₀).

δ _Pn ::

Haugh (2006) estimates the range for the fibroblast consumption of PDGF to be 2.4<δ _Pn <48/day, and proposes that a reasonable value for this parameter is 2.4/day, a value which Monine and Haugh (2008) also adopts. After accounting for the cell density, we obtain an estimate for fibroblast PDGF consumption of δ _Pn =2.4 cm³/cell/day

a _ρβ ::

Eickelberg et al. (1999) found a 2–3-fold increase in collagen expression by human lung fibroblasts in the presence of TGFβ. We assume that TGFβ induces a similar increase in collagen production by dermal fibroblasts. Hence, we estimate that a _ρβ =2/β ₀.

a _z ::

Oono et al. (2002) estimates the collagenase accumulation over one day to be 5–35 ng/mL. Using this value, and the steady state values for collagen density (∼15 μg/mg, Dale et al. 1996), fibroblasts (r/θ _nn ), collagenase (∼0.1 ng/mL, determined from Dale et al. 1996) and recognizing that the velocity, myofibroblast density and TGFβ concentration are zero, we may substitute into (16) and determine a value for collagenase production. We estimate its value to be a _z =3.37×10⁻⁹ ng/cell/day.

b _z ::

Overall et al. (1991) found a reduction of 66–75% of collagenase synthesis in the presence of TGFβ. This gives an estimate of b _z =3/β ₀.

δ _z ::

Overall et al. (1991) estimate the half-life of MMP-2 as 46 hours. We assume that collagenase (MMP-1) has the same half-life, giving a decay rate of 0.3616/day.

s::

Following Tranquillo and Murray (1992), Olsen et al. (1995) and Javierre et al. (2009), we consider a tethering coefficient of s=1.

μ::

We follow Olsen et al. (1995) and Javierre et al. (2009), and choose μ such that its non-dimensional value is 20.

E::

Estimates of E range from 1–300 N/cm² (Silver et al. 2001; Genzer and Groenewold 2006). We consider an area of approximately 1 cm², which gives a range of E of 10<E<300 N. We use the lower limit, such that E=10 N.

τ::

In Murphy et al. (2011), we estimated a range for τ of 1<τ<3 μN/cell. Hence, we consider a value of τ=2.65 μN/cell, consistent with Fray et al. (1998) and Wrobel et al. (2002).

ξ::

Wrobel et al. (2002) found that myofibroblasts can apply up to twice the cell traction force generated by fibroblasts. Hence, we choose ξ=2.