Epidermal Wound Healing: A Mathematical Model

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1 Epidermal Wound Healing: A Mathematical Model Abi Martinez, Sirena Van Epp, Jesse Kreger April 16,

2 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 2

3 1 Wound Healing History Many different methods for treating epidermal wounds have been recorded throughout history. Records of deep wounds that do not reach the bones have been documented in ancient Egypt as early as 2500 B.C. Both the descriptions and treatments of such wounds are spelled out in hieroglyphics. In ancient Rome, Emperor Nero (37 A.D to 68 A.D.), is said to have had his mother Agrippina murdered by stabbing. Nero then proceeded to examine her corpse over a glass of wine. One of the most influential and advanced surgeons, Galen (129 A.D-216 A.D.) helped pave the way for anatomy and physiology to develop. Galen would dissect corpses and study the 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 four elements, fire, earth, water, and air. Galen thought that these four elements were manifested in the human body as yellow bile, black bile, water, and phlegm respectively. Galen wrote about all of his findings and theories in four books, which anyone with even the most basic medical 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 great Galen. However, Henri de Mondeville who lived during the Middle Ages threw out that notion stating that God did not use all of his powers on Galen. Mondeville completely changed the surgical process in the Middle Ages. His methods for addressing wounds were much more modern than Galen s processes. Mondeville s methods included cleaning a wound without probing it, dressing it with nonirritating dressings, and closing the wound for faster healing. He also developed other surgical methods including a method for replacing lost virginity. Mondeville s ideas really paved the way for a more modern understanding of the human body. 3

4 2 Biology of Epidermal Wound Healing The epidermis is the outermost layer of cells in the skin. The epidermis is extremely thin at an average of mm thick. Later in our model, we will approximate the epidermis/wound to be two dimensional. This is because the diameter of the wound will be much larger relative to the mm thickness of the skin. Epidermal wound healing consists of three main stages, inflammation, wound closing, and remodeling. Before the first stage starts, platelets (blood cells) and fibrin (protein involved with the clotting of the blood) gather at the wound and clot together so that the wound does not continue to bleed. Platelets are able to aggregate in 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 continue to bleed without anything stopping it. Once this processes is complete, a signal is released to start the stages of healing. During inflammation, the platelets release a number of substances, one of them being a soluble solution, pus that contains PMNs which are a type of phagocyte. Phagocytes are cells that ingest the debris and bacteria in the wound. They are the first responders to the wound. Once the PMNs have had a chance to do their job, macrophages move into 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 with the use of activators and inhibitors. (Mitosis is the division of cells, essentially how cells reproduce). Other substances released by the platelets increase cell migration to the wound area. The main goal of inflammation is to remove any harmful substances from the wound and prepare the wound for full healing. However, if inflammation lasts too long, the tissue near 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 the platelets have had the chance to form over the wound and protect it from the environment around it. Towards the end of the inflammation stage and the beginning of wound closing, fibroblasts move 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 down collagen, which is what strengthens the area around the wound replacing the clot formed by platelets. During migration, some of the cells spread across the wound while the edges of the wound contract. At first, there is no immediate increase of cell mitosis and the cells continue 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 normal rate. It is important to understand that the wound area becomes a barren surface area but the other areas around the wound stay at normal conditions. It is only on the edges of the wound where anything happens. Once the cells around the wound have multiplied, the new cells/epidermis can migrate over the wound to close it. Once stage two is complete, the purpose 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 tension lines, i.e. rearranged to look like the skin that was there before the wound. Basically, a new skin layer forms over the healed wound returning the area back to its original state. 4

5 The biology section of epidermal wound healing will conclude with the difference between the 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, increased levels of the activator chemical are released to catalyze mitotic generation (increase the rate of mitosis). However, cell reproduction cannot indefinitely increase, which is why the inhibitory chemical is necessary. The inhibitory chemical completely shuts off a cell s ability to perform mitosis. As the chemical moves around, it slowly affects more cells. 5

6 3 Mathematical Model of Epidermal Wound Healing We begin our model based on the biology and a model that according to Sherratt and Murray has tested well with experiments. It is based on two conservation equations, one for cell density and one for concentration of the mitosis regulating chemical. It is impossible to write just one differential equation that is a successful approximation of an epidermal wound healing model. The simplest model that can be made requires the two differential equations below. 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. Note that cell density is n and the chemical concentration is c. The first term of (1) which models the cell migration is modeled by D 2 n. D is a diffusion constant and 2 is the Laplacian operator commonly used to model diffusion and cell migration. This can be represented mathematically as follows: D 2 n = D( )n ( δ = D δx + δ ) ( δ δy + ( ) δ 2 = D δx + δ2 2 δy + n 2 = D(n xx + n yy + ) = Dn xx + Dn yy + δx + δ δy + where x, y,... are the spatial variables. In this paper we are approximating the wounds as 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 complicated biological model. To begin to understand this term, Sherratt and Murray introduced s(c) which is a function of chemical concentration c. We have that s(c) is quantitatively different based on the activator or inhibitor chemical. In the unwounded condition, represented by c 0 and n 0, we need that s(c 0 ) = k where k is the linear mitotic rate. This is because in the unwounded state we have that mitotic generation - ) ) natural loss =kn (2 nn0 kn = kn (1 nn0 which is in the form of the logistic 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. 6 ) n

7 The third term of (1) which models the natural loss of cells is needed because epidermal skin cells are constantly shedding. This term will be proportional to n, so it is modeled with kn where k is a positive parameter. The first term of (2) which models the diffusion of the chemical concentration, is modeled by D c 2 c. D c is a diffusion constant and 2 is the Laplacian operator commonly used to model diffusion and cell migration. Similarly to above, we have that D c 2 c = Dc xx + Dc yy. The second term of (2) which models the production of c (c is the mitosis regulating chemical) depends on whether c activates or inhibits mitosis. We will model the production 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 the unwounded condition there is no chemical in the first place, and thus f(n 0 ) = λc 0 to cancel out the decay of the active chemical. We have that: for the activator of mitosis and f(n) = λc 0 n n 0 ( ) n α 2 n 2 + α 2 for the inhibitor of mitosis. f(n) = λc 0 n 0 In both models, we have f(0) = 0 and f(n 0 ) = λc 0 as desired. We have also introduced the 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 shown using ( basic calculus. ) We ( have that ) the derivate ( of f) with ( respect to n is: f (n) = 1 n 2 λc 0 +α 2 n o n 0 + λc n 2 +α 2 0 n 0 n2 0 +α2 1 n 2 2n = λc 0 +α 2 (n 2 +α 2 ) 2 o n 0 1 ). 2n2 n 2 +α 2 n 2 +α 2 ( Setting this equal to 0 we obtain that 1 2n2 = 0 and then α 2 n 2 = 0, and n n 2 +α 2 ) 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 the laws 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 2 n + s(c) (2 nn0 kn (3) δc δt = D c 2 c + f(n) λc (4) with initial conditions n = c = 0 at t = 0 inside the wound domain and boundary conditions n = n 0, c = c 0 on the wound boundary for all t. Please note that in this paper we will be 7

8 considering epidermal wounds that are circular in shape. 8

9 4 Numerical Solution for the Epidermal Wound Repair Model Sherratt and Murray solved the system numerically in a radially symmetric geometry. The following figure illustrates the solutions along with experimental values Sherratt and Murray acquired. 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 the inhibitor mechanism. Note, this is the case with no epidermal contraction and the model solutions have wounds of one centimeter diameters. The values for the activator mechanism are as follows: D = D c = 0.45 λ = 30 α = 0.1 9

10 Those for the inhibitor mechanism are: D = 10 4 D c = 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. Sherratt and 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 traveling wave forms (elaborated in following section). Sherratt and Murray also plotted n and c against r over equally spaced times. This can be seen 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 a wound radius of 0.5cm, the dimensional wavespeeds are mmh 1 for the activator and mmh 1 for the inhibitor. Comparing the two solutions demonstrates very little difference between the inhibitor and activator mechanisms. However, there is a difference in wound healing time in the two mechanisms due to the traveling wave solutions discussed in the next section. 10

11 Figure 2: Cell density n and chemical concentration c as a function of radius r at a selection of equally spaced times. (a) Biochemical activation of mitosis with parameter values D = , D c = 0.45, λ = 30, α = 0.1; (b) biochemical inhibition of mitosis with parameter D = 10 4, D c = 0.85, λ = 5. (From Sherratt and Murray 1990, 1991) 11

12 5 Simplified System of Ordinary Differential Equations In this section, Murray considers the possibility of traveling waves solutions to the partial differential equations model given by equations (3) and (4). To simplify the model we will assume that z = x + at where a is the wavespeed. We now have that n(x, t) = N(z) and c(x, t) = C(z) and thus the partial differential equations model becomes a coupled system of ordinary differential equations given by: an = DN + s(c)n(2 N) N (5) ac = D c C + 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 of D = 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: with the same initial conditions. N = N a + 1 s(c)n(2 N) (7) a C = a D c C + λ D c C λ D c g(n) (8) Murray and Sherratt solved this system again using numerical methods including phase space analysis and a regular perturbation method. They found that the solutions for both the activator and the inhibitor closely with the numerical solution to the system of partial differential equations in the previous section. Thus they concluded that the approximations were 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 against each other. The top two graphs correspond to the model for the activator. The independent variable is the radius of the wound and the dependent variable is the cell density and chemical concentration respectively. Similarly, the bottom two graphs represent the model for the inhibitor. The independent variable is the radius of the wound and the dependent variable is the cell density and the chemical concentration respectively. The solid curve in each graph is the numerical solution to the partial differential equation model, whereas the dotted curve is the numerical solution to the simplified ordinary differential equation model with the approximations outlined in this section. As we can see from analysis of the graphs, the two different solution techniques generally match up very closely. 12

13 Figure 3: Comparison of Different Numerical Solutions 13

14 6 Clinical Implications Our previous model focuses on the chemical autoregulation of cell division. However, Sherratt and Murray also wanted to investigate the effect of applying additional quantities of mitosisregulating chemicals. Originally, they added a chemical concentration to the wound so high that they believed the experiment would be unrealistic. They found that this had no affect on the wound whatsoever. They then experimented by gradually releasing different rates of regulatory chemicals into the wounds by applying a dressing over the wound. By doing this, they were adding a constant term c dress to the right-hand side of the c-equation. The results are shown in Figure 4 below: Figure 4: Epidermal Wound Healing Though the results weren t fully quantitative and mostly qualitative, the results of the gradually released chemicals still showed significant results. The results showed that the higher c dress was, the faster the wound would heal. Basically, the faster the mitosis-regulating chemicals were released, the time it took for activators to be released decreased and the time it took for inhibitors to be released increased. The cells were signaled to start mitosis earlier and told to stop dividing later. For simplicity, only circular wounds were considered above. However, the model can be applied to any initial wound. Actually one of the original aims of the model was to see if and how the shape of a wound affected healing. The following function represents the boundaries of cusp-ovate wounds: 14

f shape (x, α) = 1 2 where 1 < α < 1. ( 1 + 1 ) [ 1 sign(α) (1 + 1α ) ( x + 1 α 2 2 2α 1 ) ] 1 2 2 2 Results showed that as α 1, the wound would be cusp shaped.

15 f shape (x, α) = 1 2 where 1 < α < 1. ( ) [ 1 sign(α) (1 + 1α ) ( x + 1 α 2 2 2α 1 ) ] Results showed that as α 1, the wound would be cusp shaped. When α = 0 the wound was a diamond, and when α > 0 the wound was oval shaped. Figure 5 shows different examples of different wounds shapes based on their α value. The different values of α, and therefore the different shapes of wounds, and the affect on healing time can be seen in Figure 6. 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. Going back to the c dress experiment from above, wounds healed faster when the activators were signaled sooner and inhibitors were signaled later. Looking at Figure 6 we can see that the value 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 activators and inhibitors grows, therefore speeding up the time of healing. So again, when α > 0 and α 1, the wound is oval or cusp shaped and according to Figure 6 is the fastest healing shaped wound. Figure 5: Noncircular Epidermal Wounds 15

16 Figure 6: Healing time vs. α value References [1] Boylan, Michael. Internet Encyclopedia of Philosophy. Galen. Internet Encyclopedia of Philosophy, n.d. Web. 08 Apr [2] D Epiro, Peter. Second Century. The Book of Firsts: 150 World-Changing People and Events, from Caesar Augustus to the Internet. New York: Anchor, Print. [3] Epidermal Wound Healing. Boundless, n.d. Web. 29 Mar epidermal-wound-healing/. [4] Galen. Galen. University of Dayton, n.d. Web. 08 Apr hume/galen/galen.htm. [5] Murray, J. D. Spatial Models and Biomedical Applications 2 2. Spatial Models and Biomedical Applications 2 2. N.p., n.d. Web. [6] Orgil, Denis, and Carlos Blanco. Biomaterials for Treating Skin Loss. Florida: Woodhead Publishing Limited, Print. 16

On mathematical modeling of epidermal wound healing

On mathematical modeling of epidermal wound healing Jesse Kreger (Occidental College) November 16, 2015 Abstract In this paper we investigate two mathematical models of epidermal wound healing. The initial