Epidermal Wound Healing: A Mathematical Model

Abi Martinez, Sirena Van Epp, Jesse Kreger

April 16, 2014

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Contents

1 Wound Healing History 3

2 Biology of Epidermal Wound Healing 4

3 Mathematical Model of Epidermal Wound Healing 6

4 Numerical Solution for the Epidermal Wound Repair Model 9

5 Simplified System of Ordinary Differential Equations 12

6 Clinical Implications 14

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1 Wound Healing History

Many different methods for treating epidermal wounds have been recorded throughout his-tory. Records of deep wounds that do not reach the bones have been documented in ancientEgypt as early as 2500 B.C. Both the descriptions and treatments of such wounds are spelledout in hieroglyphics. In ancient Rome, Emperor Nero (37 A.D to 68 A.D.), is said to havehad his mother Agrippina murdered by stabbing. Nero then proceeded to examine her corpseover a glass of wine.

One of the most influential and advanced surgeons, Galen (129 A.D-216 A.D.) helpedpave the way for anatomy and physiology to develop. Galen would dissect corpses and studythe human body. He was interested in uncovering the internal organs and their functions.Galen built upon Aristotle’s pluralistic theory. That is, Aristotle claimed that there were fourelements, fire, earth, water, and air. Galen thought that these four elements were manifestedin the human body as yellow bile, black bile, water, and phlegm respectively. Galen wroteabout all of his findings and theories in four books, which anyone with even the most basicmedical background could understand.

After Galen, not many surgeons or physicians were inclined to study the body further,since they believed it would be arrogant to think they could learn more than the greatGalen. However, Henri de Mondeville who lived during the Middle Ages threw out thatnotion stating that God did not use all of his powers on Galen. Mondeville completelychanged the surgical process in the Middle Ages. His methods for addressing wounds weremuch more modern than Galen’s processes. Mondeville’s methods included cleaning a woundwithout probing it, dressing it with nonirritating dressings, and closing the wound for fasterhealing. He also developed other surgical methods including a method for replacing lostvirginity. Mondeville’s ideas really paved the way for a more modern understanding of thehuman body.

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2 Biology of Epidermal Wound Healing

The epidermis is the outermost layer of cells in the skin. The epidermis is extremely thin at anaverage of 0.775 mm thick. Later in our model, we will approximate the epidermis/wound tobe two dimensional. This is because the diameter of the wound will be much larger relative tothe 0.775 mm thickness of the skin. Epidermal wound healing consists of three main stages,inflammation, wound closing, and remodeling. Before the first stage starts, platelets (bloodcells) and fibrin (protein involved with the clotting of the blood) gather at the wound andclot together so that the wound does not continue to bleed. Platelets are able to aggregatein a certain area and stick together due to the sticky fibrin proteins on their cell membranes.A low platelet count is dangerous because without enough of them, a wound can continueto bleed without anything stopping it. Once this processes is complete, a signal is releasedto start the stages of healing.

During inflammation, the platelets release a number of substances, one of them beinga soluble solution, pus that contains PMNs which are a type of phagocyte. Phagocytesare cells that ingest the debris and bacteria in the wound. They are the first respondersto the wound. Once the PMNs have had a chance to do their job, macrophages moveinto the site to clear up the used phagocytes. The macrophages also secret growth factors.Growth factors stimulate the cells around the wound to increase their rate of mitosis withthe use of activators and inhibitors. (Mitosis is the division of cells, essentially how cellsreproduce). Other substances released by the platelets increase cell migration to the woundarea. The main goal of inflammation is to remove any harmful substances from the woundand prepare the wound for full healing. However, if inflammation lasts too long, the tissuenear and around the wound can become damaged. As long as there is debris in the wound,inflammation will continue to run. This is why it is important to keep a wound clean until theplatelets have had the chance to form over the wound and protect it from the environmentaround it.

Towards the end of the inflammation stage and the beginning of wound closing, fibroblastsmove into the wound site. Fibroblasts are the most common connective tissues in animals.These are the cells that will migrate and multiply to heal the wound. They lay downcollagen, which is what strengthens the area around the wound replacing the clot formedby platelets. During migration, some of the cells spread across the wound while the edgesof the wound contract. At first, there is no immediate increase of cell mitosis and the cellscontinue to divide at a normal rate. Once the cells are done migrating to the site however,then the mitotic activity increases at the edges of the wound to about 15 times the normalrate. It is important to understand that the wound area becomes a barren surface area butthe other areas around the wound stay at normal conditions. It is only on the edges ofthe wound where anything happens. Once the cells around the wound have multiplied, thenew cells/epidermis can migrate over the wound to close it. Once stage two is complete, thepurpose of remodeling is to fix the originally disorganized healing that happened in stage two.The collagen that was originally hastily laid down is rearranged and aligned along tensionlines, i.e. rearranged to look like the skin that was there before the wound. Basically, a newskin layer forms over the healed wound returning the area back to its original state.

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The biology section of epidermal wound healing will conclude with the difference betweenthe activator and inhibitor chemicals, which will be the most important later in the paper.When an epidermal wound occurs and the cells migrate to surround the wound, increasedlevels of the activator chemical are released to catalyze mitotic generation (increase therate of mitosis). However, cell reproduction cannot indefinitely increase, which is why theinhibitory chemical is necessary. The inhibitory chemical completely shuts off a cell’s abilityto perform mitosis. As the chemical moves around, it slowly affects more cells.

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3 Mathematical Model of Epidermal Wound Healing

We begin our model based on the biology and a model that according to Sherratt andMurray has tested well with experiments. It is based on two conservation equations, one forcell density and one for concentration of the mitosis regulating chemical. It is impossible towrite just one differential equation that is a successful approximation of an epidermal woundhealing model. The simplest model that can be made requires the two differential equationsbelow.

These two starting equations in word form are:

rate of change of cell density, n = cell migration + mitotic generation − natural loss

(1)

rate of increase of chemical concentration, c

= diffusion of c + production of c by cells − decay of active chemical (2)

To create the mathematical model, consider what each of those terms will look like. Notethat cell density is n and the chemical concentration is c.

• The first term of (1) which models the cell migration is modeled by D∇2n. D is adiffusion constant and ∇2 is the Laplacian operator commonly used to model diffusionand cell migration. This can be represented mathematically as follows:

D∇2n = D(∇ · ∇)n

= D

(δ

δx+

δ

δy+ · · ·

)·(δ

δx+

δ

δy+ · · ·

)n

= D

(δ2

δx2+

δ2

δy2+ · · ·

)n

= D(nxx + nyy + · · · )= Dnxx +Dnyy + · · ·

where x, y, ... are the spatial variables. In this paper we are approximating the woundsas two dimensional shapes and thus we will only have two spatial variables.

• The second term of (1) which models the mitotic generation of cells is a very com-plicated biological model. To begin to understand this term, Sherratt and Murrayintroduced s(c) which is a function of chemical concentration c. We have that s(c) isquantitatively different based on the activator or inhibitor chemical. In the unwoundedcondition, represented by c0 and n0, we need that s(c0) = k where k is the linear mi-totic rate. This is because in the unwounded state we have that mitotic generation -

natural loss =kn(

2− nn0

)− kn = kn

(1− n

n0

)which is in the form of the logis-

tic growth model as desired. So we will model the mitotic generation term with

s(c)n(

2− nn0

)where s(c) depends on the activator or the inhibitor chemical.

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• The third term of (1) which models the natural loss of cells is needed because epidermalskin cells are constantly shedding. This term will be proportional to n, so it is modeledwith kn where k is a positive parameter.

• The first term of (2) which models the diffusion of the chemical concentration, ismodeled by Dc∇2c. Dc is a diffusion constant and ∇2 is the Laplacian operator com-monly used to model diffusion and cell migration. Similarly to above, we have thatDc∇2c = Dcxx +Dcyy.

• The second term of (2) which models the production of c (c is the mitosis regulat-ing chemical) depends on whether c activates or inhibits mitosis. We will model theproduction of c by the function f(n). The function f(n) must satisfy the properties:with no cells there will be no production of c and thus f(0) = 0 and also that in theunwounded condition there is no chemical in the first place, and thus f(n0) = λc0 tocancel out the decay of the active chemical. We have that:

f(n) = λc0 ·n

n0

·(n20 + α2

n2 + α2

)for the activator of mitosis and

f(n) =λc0n0

· n

for the inhibitor of mitosis.

In both models, we have f(0) = 0 and f(n0) = λc0 as desired. We have also introducedthe positive parameter α, which relates to the maximum rate of chemical production.This is because when n = α, f(n) will achieve its maximum. This can be shownusing basic calculus. We have that the derivate of f with respect to n is: f ′(n) =

λco1n0

(n20+α

2

n2+α2

)+ λc0

nn0

(− n2

0+α2

(n2+α2)2

)2n = λco

1n0

(n20+α

2

n2+α2

)(1− 2n2

n2+α2

).

Setting this equal to 0 we obtain that(

1− 2n2

n2+α2

)= 0 and then α2 − n2 = 0, and

thus α = n is a critical value. Using the first derivative test we can see that (α, f(α))is indeed a maximum.

• The third term of (2) which models the decay of the active chemical should follow thelaws of first order kinetics and thus look like −λc where λ is a positive rate constant.

Now that we have all of the terms, we can represent the two main equations mathematically.They are as follows:

δn

δt= D∇2n+ s(c)

(2− n

n0

)− kn (3)

δc

δt= Dc∇2c+ f(n)− λc (4)

with initial conditions n = c = 0 at t = 0 inside the wound domain and boundary conditionsn = n0, c = c0 on the wound boundary for all t. Please note that in this paper we will be

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considering epidermal wounds that are circular in shape.

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4 Numerical Solution for the Epidermal Wound Repair

Model

Sherratt and Murray solved the system numerically in a radially symmetric geometry. Thefollowing figure illustrates the solutions along with experimental values Sherratt and Murrayacquired.

Figure 1: The decrease in wound radius with time for normal healing of a circular wound,with time expressed as a percentage of total healing time

In Figure 1, the solid line denotes the activator mechanism and the dotted line denotes theinhibitor mechanism. Note, this is the case with no epidermal contraction and the modelsolutions have wounds of one centimeter diameters. The values for the activator mechanismare as follows:

• D = 5× 10−4

• Dc = 0.45

• λ = 30

• α = 0.1

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Those for the inhibitor mechanism are:

• D = 10−4

• Dc = 0.85

• λ = 5

The points shown in Figure 1 represent values obtained during several different experiments.As you can see, the solutions compare well with the experimental data.

In the solutions, the change in wound radius with respect to time was recorded. Sherrattand Murray considered the wound healed when the cell density reached 80%. Although 80%is an arbitrary number, it will not significantly alter the results since solutions have travelingwave forms (elaborated in following section).

Sherratt and Murray also plotted n and c against r over equally spaced times. This can beseen in Figure 2. These solutions demonstrate two phases, a lag phase and a linear phase.The speed of the linear phase can be calculated from Figure 2 below. For example, for awound radius of 0.5cm, the dimensional wavespeeds are 2.6× 10−3mmh−1 for the activatorand 1.2× 10−3mmh−1 for the inhibitor.

Comparing the two solutions demonstrates very little difference between the inhibitor andactivator mechanisms. However, there is a difference in wound healing time in the twomechanisms due to the traveling wave solutions discussed in the next section.

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Figure 2: Cell density n and chemical concentration c as a function of radius r at a selectionof equally spaced times. (a) Biochemical activation of mitosis with parameter values D =5 · 10−4, Dc = 0.45, λ = 30, α = 0.1; (b) biochemical inhibition of mitosis with parameterD = 10−4, Dc = 0.85, λ = 5. (From Sherratt and Murray 1990, 1991)

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5 Simplified System of Ordinary Differential Equations

In this section, Murray considers the possibility of traveling waves solutions to the partialdifferential equations model given by equations (3) and (4). To simplify the model we willassume that z = x + at where a is the wavespeed. We now have that n(x, t) = N(z) andc(x, t) = C(z) and thus the partial differential equations model becomes a coupled systemof ordinary differential equations given by:

aN ′ = DN ′′ + s(C)N(2−N)−N (5)

aC ′ = DcC′′ + λg(N)− λC (6)

with biologically appropriate initial conditions of N(−∞) = C(−∞) = 0, N(∞) = C(∞) =1, N ′(±∞) = C ′(±∞) = 0.

To analyze this system of ordinary differential equations, consider the approximation ofD = 0. This seems like a reasonable approximation as it was previously found that D ≤ 10−4

for both the activator and inhibitor. This reduces the system to:

N ′ = −Na

+1

as(C)N(2−N) (7)

C ′′ =a

Dc

C ′ +λ

Dc

C − λ

Dc

g(N) (8)

with the same initial conditions.

Murray and Sherratt solved this system again using numerical methods including phasespace analysis and a regular perturbation method. They found that the solutions for boththe activator and the inhibitor closely with the numerical solution to the system of partialdifferential equations in the previous section. Thus they concluded that the approximationswere both helpful and safe approximations to use to simplify the mathematics involved.

Figure 3 on the following page are plots of how the numerical solutions stacked up againsteach other. The top two graphs correspond to the model for the activator. The independentvariable is the radius of the wound and the dependent variable is the cell density and chemicalconcentration respectively. Similarly, the bottom two graphs represent the model for theinhibitor. The independent variable is the radius of the wound and the dependent variable isthe cell density and the chemical concentration respectively. The solid curve in each graphis the numerical solution to the partial differential equation model, whereas the dotted curveis the numerical solution to the simplified ordinary differential equation model with theapproximations outlined in this section. As we can see from analysis of the graphs, the twodifferent solution techniques generally match up very closely.

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Figure 3: Comparison of Different Numerical Solutions

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6 Clinical Implications

Our previous model focuses on the chemical autoregulation of cell division. However, Sherrattand Murray also wanted to investigate the effect of applying additional quantities of mitosis-regulating chemicals. Originally, they added a chemical concentration to the wound so highthat they believed the experiment would be unrealistic. They found that this had no affecton the wound whatsoever. They then experimented by gradually releasing different rates ofregulatory chemicals into the wounds by applying a dressing over the wound. By doing this,they were adding a constant term cdress to the right-hand side of the c-equation. The resultsare shown in Figure 4 below:

Figure 4: Epidermal Wound Healing

Though the results weren’t fully quantitative and mostly qualitative, the results of the grad-ually released chemicals still showed significant results. The results showed that the highercdress was, the faster the wound would heal. Basically, the faster the mitosis-regulating chem-icals were released, the time it took for activators to be released decreased and the time ittook for inhibitors to be released increased. The cells were signaled to start mitosis earlierand told to stop dividing later.

For simplicity, only circular wounds were considered above. However, the model can beapplied to any initial wound. Actually one of the original aims of the model was to see if andhow the shape of a wound affected healing. The following function represents the boundariesof cusp-ovate wounds:

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fshape(x, α) =1

2

(1 +

1

α

)− sign(α)

[1

2

(1 +

1

α2

)−(x+

1

2α− 1

2

)2] 1

2

where −1 < α < 1.

Results showed that as α → 1, the wound would be cusp shaped. When α = 0 the woundwas a diamond, and when α > 0 the wound was oval shaped. Figure 5 shows differentexamples of different wounds shapes based on their α value. The different values of α, andtherefore the different shapes of wounds, and the affect on healing time can be seen in Figure6. It seems that the most ideal shape of wound is an oval or cusp because it heals the fastest.We can may come to this conclusion by evaluating the information given in Figure 6. Goingback to the cdress experiment from above, wounds healed faster when the activators weresignaled sooner and inhibitors were signaled later. Looking at Figure 6 we can see that thevalue of α changes the healing time based on mitosis-regulating activators and inhibitors.When the value of α is greater than zero and approaches 1, the difference between activatorsand inhibitors grows, therefore speeding up the time of healing. So again, when α > 0 andα → 1, the wound is oval or cusp shaped and according to Figure 6 is the fastest healingshaped wound.

Figure 5: Noncircular Epidermal Wounds

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Figure 6: Healing time vs. α value

References

[1] Boylan, Michael. “Internet Encyclopedia of Philosophy.” Galen. Internet Encyclopediaof Philosophy, n.d. Web. 08 Apr. 2014. http://www.iep.utm.edu/galen/.

[2] D’Epiro, Peter. “Second Century.” The Book of Firsts: 150 World-Changing People andEvents, from Caesar Augustus to the Internet. New York: Anchor, 2010. 41-43. Print.

[3] Epidermal Wound Healing. Boundless, n.d. Web. 29 Mar. 2014.https://www.boundless.com/physiology/the-integumentary-system/wound-healing/epidermal-wound-healing/.

[4] “Galen.” Galen. University of Dayton, n.d. Web. 08 Apr. 2014.http://campus.udayton.edu/ hume/Galen/galen.htm.

[5] Murray, J. D. “Spatial Models and Biomedical Applications 2 2.” Spatial Models andBiomedical Applications 2 2. N.p., n.d. Web. http://site.ebrary.com/id/10047723.

[6] Orgil, Denis, and Carlos Blanco. Biomaterials for Treating Skin Loss. Florida: WoodheadPublishing Limited, 2009. Print.

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