A mathematical model of tissue replacement duringepidermal wound healing

Sophia A. Maggelakis *

Department of Mathematics and Statistics, Rochester Institute of Technology, Rochester, NY 14623, USA

Received 2 January 2001; accepted 3 September 2002

Abstract

A mathematical model, which describes the control of the development and growth of a healing unit, is

presented. The replacement of epidermal injured tissue, which is controlled by a negative feedback

mechanism, is modeled in one-dimensional geometry. The model is based on diffusion equations that relate

the production of macrophage-derived growth factors (MDGFs) to oxygen availability, the capillary

density growth to MDGF production and concentration, and the oxygen concentration to the growth ofcapillary density. The results of the model suggest that the normal healing of a circular epidermal wound

depends on the oxygen supply, and in order for successful healing to take place, the oxygen concentration

within the wound space must be at low levels.

� 2002 Elsevier Science Inc. All rights reserved.

Keywords: Macrophage-derived growth factor; Epidermal wound healing; Neovascularization; Angiogenesis

1. Introduction

Wound healing has been most closely studied in the skin. Skin is composed of an outer layercalled the epidermis and an inner layer called the dermis. The dermis contains living cells, bloodvessels, nerves, and protein fibers, while the epidermis contains mostly dead cells which moveupward from the dermis. The simplest situation of a wound occurring on the skin is created by acut through the epidermis into the dermis or when part of the epidermis is removed [1]. Thehealing of an epidermal wound involves cell migration, cell mitosis, and inhibition of mitosis [2–5].During the process of cell migration, epidermal cells move across the wound area in an attempt to

* Fax: 1-716-457-5766.

E-mail address: sxmsma@rit.edu (S.A. Maggelakis).

0307-904X/02/$ - see front matter � 2002 Elsevier Science Inc. All rights reserved.

PII: S0307-904X(02)00100-2

Applied Mathematical Modelling 27 (2003) 189–196

www.elsevier.com/locate/apm

re-establish the continuity of the epidermis. This is followed by a burst of mitotic activity whichprovides an additional population of cells and contributes to the thickness of the epidermis. Oncethe appropriate thickness has been reached, there is inhibition of mitosis to prevent the formationof more epidermal cells.

A wounded area is prepared for healing by the removal of all contaminants (phagocytes).Keeping the wound decontaminated, however, is not enough for the healing to proceed. Growthof new blood vessels (neovascularization) is necessary to supply the damaged tissue with oxygenand nutrients [6]. Experiments have shown that the development and growth of a healing unit arecontrolled by a negative feedback mechanism [3], which includes the oxygen concentration withinthe cluster tissue and wound space and various macrophage-derived growth factors (MDGFs)[7–9]. When the concentration of oxygen is at low levels, macrophages appear at the wound site.These macrophages release chemical substances (MDGFs) such as vascular endothelial growthfactor and transforming growth factor-b that have the capacity to stimulate vessel growth andcollagen deposition. MDGFs trigger the endothelial cells of the nearby blood vessels causing themto participate in the process. Many experiments have suggested that the generation of new bloodvessels within the wounded area is stimulated and maintained as long as such growth factors arepresent. The newly formed capillaries transfer oxygen and vital nutrients to those cells in theinjured tissue which are involved in the repairing process. Experimental results point out thatinsufficient blood supply affects the healing process as a whole, and that the rate of wound healingdepends on the oxygen supply [10]. Thus, for successful healing to take place, the oxygen levelwithin the wound space must be low.

A number of mathematical models, that investigate certain aspects of the complicated processof wound healing, exist in the literature [11–15]. In this paper, a mathematical model is developedwhich is based on diffusion equations to describe the dependence of tissue regeneration on oxygenavailability, production of MDGFs, and the growth of capillary density.

2. Model development

2.1. Oxygen transport

Since the thickness of adult skin is approximately 1–2 mm thick, it is safe to regard the skin astwo-dimensional. The case of a circular wound is thus considered, which is described in terms ofaxisymmetric geometry using one spatial coordinate. The center of the wound is taken to be atx ¼ 0 and the radius at x ¼ R. The distance from the center of the wound is denoted by x.

When a wound occurs, blood vessels that extend across the injured tissue are cut, and blood ispoured into the wound. The blood coagulates inside the wound, closes up the blood vessels thathave been injured, and in turn temporarily closes the wound. A wounded area is prepared forhealing by the removal of all contaminants (phagocytosis), and by the formation of new bloodvessels (neovascularization) that will supply the damaged tissue with new blood and as a resultwith oxygen and nutrients. Insufficient blood supply, therefore, can affect the healing process as awhole. It is assumed that the capillary tips act as the main source of oxygen so that the woundspace is supplied with oxygen by the new blood vessels. If Co is used to denote the oxygen con-centration in the wounded area, and nðx; tÞ is used to represent the capillary density at position x

190 S.A. Maggelakis / Appl. Math. Modelling 27 (2003) 189–196

and time t, the following diffusion equation is used to model the motion of oxygen through thewound space

oCo

ot¼ Dor2Co þ F ðnÞ � GðCoÞ; 06 x6R ð1Þ

where Do is the diffusion coefficient. The function F ðnÞ acts as a source term to represent thesupply of oxygen to the damaged tissue by the capillaries, and it is chosen to be F ðnÞ ¼ knnðx; tÞ,with kn being the oxygen production rate. The consumption of oxygen by the MDGFs is describedby a sink term, which is taken to be GðCoÞ ¼ koðCo=ChÞ, where ko represents the oxygen con-sumption rate, and Ch represents the critical level of oxygen concentration below which macro-phages appear at the wound site.

Diffusive equilibrium is assumed so that the above equation takes the form

Dor2Co þ knnðx; tÞ � ko

Co

Ch¼ 0; 06 x6R ð2Þ

It is assumed that oxygen is not diffused beyond the center of the wound so that dCo=dx ¼ 0 atx ¼ 0 and that CoðRÞ ¼ Ci where Ci represents the oxygen concentration at the periphery of thecircular wound. Solving the above differential equation subject to the stated boundary conditionsyields

CoðxÞ ¼ Ci

�� HnðxÞ

K

�coshð

ffiffiffiffiK

pxÞ

coshðffiffiffiffiK

pRÞ

þ HnðxÞK

; 06 x6R: ð3Þ

2.2. Modeling the action of macrophage-derived growth factor

When the concentration of oxygen is at low levels, macrophages appear at the wound site.These macrophages release chemical substances referred to as MDGFs that have the capacity tostimulate vessel growth and collagen deposition. MDGFs trigger the endothelial cells of thenearby blood vessels causing them to participate in the process. Linear diffusion is used to describethe motion of MDGFs that are produced by the macrophages, and the dependence of growthfactor production on oxygen concentration is taken into account. The production, depletion andabsorption of MDGFs are described by the following system:

oCm

ot¼ Dmr2Cm þ kmQðCoÞ � kCm; 06 x6R ð4Þ

and

oCm

ot¼ Dmr2Cm � kCm � kc

Cm

Cmax

; R6 x6Rmax ð5Þ

where Cm denotes the concentration of MDGFs and Dm is the diffusion coefficient. The para-meters km, k, and kc are used to represent the production rate of MDGFs by the macrophages, thenatural loss or depletion rate of MDGFs within and outside the wound, and the consumptionrate of MDGFs by the capillary tips respectively. The source function QðCoÞ, which is modeledby QðCoÞ ¼ 1 � ðCo=ChÞ, is used to describe the dependence of MDGFs on oxygen, and the

S.A. Maggelakis / Appl. Math. Modelling 27 (2003) 189–196 191

parameter Cmax is used to represent the threshold level below which the concentration of MDGFsis not enough to stimulate endothelial cell proliferation.

Angiogenesis, a process by which new blood vessels are generated through sprouting fromalready existing surrounding blood vessels, begins outside the wound area causing the capilla-ries to move towards the wound site. After the injury, the endothelial cells at the boundary of thewound begin to regenerate forming a shell on the perimeter of the wound [16], and thenewly generated endothelial cells begin to migrate as long as good blood supply is available. Itis assumed that MDGFs are carried out from the wound space by the blood and away fromthe wound site by the vasculature, with Rmax being the farthest extend of MDGFs. The addi-tional boundary conditions, therefore, that are applied to solve the above system at steady state,are that

dCm

dx¼ 0 at x ¼ 0;Cm and

dCm

dxare continuous at x ¼ R; and

dCm

dx¼ 0 at x ¼ Rmax

so that

CmðxÞ ¼ ðC1 þ V1Þ coshðmxÞ þ V2 sinhðmxÞ; 06 x6R ð6Þ

and

CmðxÞ ¼ C3 coshðwxÞ þ C4 sinhðwxÞ; R6 x6Rmax ð7Þwhere m ¼ ðk=DmÞ1=2

, w ¼ ððk=DmÞ þ ðkc=DmCmaxÞÞ1=2, and

V1 ¼ � A1

m2coshðmxÞ � A2

2mðlþ mÞ coshðlþ mÞxþ A2

2mðl� mÞ coshðl� mÞx ð8Þ

V2 ¼A1

m2sinhðmxÞ þ A2

2mðlþ mÞ sinhðlþ mÞxþ A2

2mðl� mÞ sinhðl� mÞx ð9Þ

with A1 ¼ �ðkm=DmÞð1 � ðHn=ChKÞÞ, A2 ¼ ðkm=DmCh coshðlRÞÞðCi � ðHn=KÞÞ, and l ¼ K1=2.

C1 ¼1

E

�� w2U sinhwðR� RmaxÞ þ Vw coshwðR� RmaxÞ

�ð10Þ

C3 ¼1

EwU coshðwRmaxÞ½ V coshðmRÞ½ � Um sinhðmRÞ ð11Þ

C4 ¼1

E½ � w sinhðwRmaxÞ V coshðmRÞ½ � Um sinhðmRÞ ð12Þ

where

E ¼ �w2 coshðmRÞ sinhwðR� RmaxÞ þ mw sinhðmRÞ coshwðR� RmaxÞ ð13Þ

U ¼ �V1ðRÞ coshðmRÞ � V2ðRÞ sinhðmRÞ ð14Þ

V ¼ �V1ðRÞm sinhðmRÞ � V 01ðRÞ coshðmRÞ � V2ðRÞm coshðmRÞ � V 0

2ðRÞ sinhðmRÞ ð15Þwith V 0

1ðxÞ ¼ dV1=dx, and V 02ðxÞ ¼ dV2=dx.

192 S.A. Maggelakis / Appl. Math. Modelling 27 (2003) 189–196

2.3. Growth of capillary density

High levels of MDGFs trigger capillary growth, which, at a particular location, depends on theconcentration of MDGFs and the capillary density at that point. This assumption is based on thefact that MDGFs stimulate capillary growth, and when the capillary density is high there aremore capillary tips present to grow. The following differential equation is used to model thisgrowth:

onðx; tÞot

¼ lCmðxÞnðx; tÞ 1

�� nðx; tÞ

L

�ð16Þ

where l is a proportionality constant representing the growth rate, and L is the limiting capillarydensity. Logistic growth occurs due to normal feedback as MDGFs go to zero across the woundsurface. After the initial distribution of capillary tips, which have formed at the parent vessel, noadditional tips originate from the that parent vessel, so that at the periphery of the wound, thecapillary tip density decays with time.

Thus,

nðtÞ ¼ Lno

no þ ðL� noÞ exp½�lCmðxÞLtð17Þ

where noðx; tÞ is the capillary density distribution at time t ¼ 0.

3. Results and discussion

In this paper, a model has been developed to include the negative feedback mechanism thatcontrols the regeneration of a wound tissue. The effects of angiogenesis on oxygenation of thewound area, which in turn affects the production of the MDGFs and as a result the growth ofcapillary density, are examined. The interactions that are taking place between oxygen, theMDGFs, and the capillary density are described by Eqs. (3), (6), (7) and (17), which are coupled inthe variables CoðxÞ, CmðxÞ, noðx; tÞ. As a result, a negative feedback mechanism is built into thesystem. When the capillaries grow beyond a certain point, they contribute to their own growthretardation, i.e., negative feedback takes place, because there is more oxygen available andtherefore less MDGF production.

The parameters involved in the model were used to obtain qualitative results to describe theimportant features of the model. Results were obtained for the oxygen concentration, MDGFconcentration, and capillary growth for different values of the parameter kn, which is used tocontrol the amount of feedback in the system. The behavior of oxygen concentration, MDGFsconcentration, and growth of capillary density is shown in Fig. 1(a), (b), and (c) respectively. Inregions where the oxygen concentration is low, there are high levels of MDGFs concentrationand, consequently, high capillary density growth. Fig. 2 illustrates that an increase in the amountof feedback results in higher levels of oxygen concentration, which in turn lowers the productionof MDGFs, as shown in Fig. 3. Low oxygen levels, however, result in increased MDGF pro-duction, which stimulates capillary growth. This is indicated in Fig. 4 where it is observed thathigh levels of MDGF induce high rates of capillary growth. It should be noted that for all cases,

S.A. Maggelakis / Appl. Math. Modelling 27 (2003) 189–196 193

the initial capillary density was assumed to be uniformly distributed over the wound area. Whenthe capillary density reaches these high levels, it allows for more oxygen to be transferred to thewounded area to compensate for low oxygen across the wound. The above results, which areconsistent with experimental results [3,7–9], suggest that the production of MDGF and the re-

WoundRadius x

0 2 4 6 8 10 12 14

Oxy

gen

Con

cent

ratio

nC

o

0

2

4

6

8

10

12

14

Wound Radius x

0 2 4 6 8 10 12 14M

DG

Fs C

once

ntra

tion

Cm

0

1

2

3

4

5

6

7

8

9

Wound Radius x

0 2 4 6 8 10 12 14 16

Cap

illar

yD

ensi

tyn

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

(a) (b)

(c)

Fig. 1. (a) The behavior of oxygen concentration within the circular wound area whose radius is taken to be R ¼ 10. (b)

The behavior of the concentration of MDGFs inside and outside the circular wound area. (c) The behavior of the

capillary density nðx; tÞ inside and outside the circular wound area with initial capillary density unifromly distributed

over the damaged tissue.

194 S.A. Maggelakis / Appl. Math. Modelling 27 (2003) 189–196

sulting capillary growth depend on the amount of negative feedback present, and that steady stateis reached as long as the parameter kn is positive.

WoundRadius x

0 2 4 6 8 10

Oxy

gen

Con

cent

ratio

nC

o

0

1

2

3

n = 0.0

n = 0.5

n = 0.9λ

λ

λ

Fig. 2. Effects of oxygen production rate on oxygen concentration. An increase in the amount of feedback results in

higher levels of oxygen concentration.

Wound Radius x

0 2 4 6 8 10 12 14

MD

GF

sC

once

ntra

tion

Cm

0

2

4

6

8

10

12

n = 0.0

n = 0.05

n = 0.1

λ

λ

λ

Fig. 3. Effects of oxygen production rate on the concentration of MDGFs. An increase in the amount of feedback

results in higher levels of oxygen concentration, which in turn lowers the production of MDGFs.

S.A. Maggelakis / Appl. Math. Modelling 27 (2003) 189–196 195

Results of models, such as the one developed in this paper, can be used to give possible jus-tification to the reason of unsuccessful wound healing and can offer possible techniques that maybe used to improve our understanding of the wound healing process.

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Wound Radius x

0 2 4 6 8 10 12 14 16

Cap

illa

ryD

ensi

tyn

0.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

n = 0.0

n = 0.05

n = 0.1

λ

λ

λ

Fig. 4. Effects of oxygen production rate on the growth of capillary density. Low levels of oxygen result in increased

production levels of MDGFs, which in turn stimulates capillary growth.

196 S.A. Maggelakis / Appl. Math. Modelling 27 (2003) 189–196