A THREE DIMENSIONAL MODEL OF WOUND HEALING: ANALYSIS AND COMPUTATION AVNER FRIEDMAN BEI HU CHUAN XUE

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1 DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SERIES B Volume 17, Number 8, November 2012 Website: pp. X XX A THREE DIMENSIONAL MODEL OF WOUND HEALING: ANALYSIS AND COMPUTATION AVNER FRIEDMAN Department of Mathematics and Mathematical Biosciences Institute Ohio State University, Columbus, OH 43210, USA BEI HU Dept of Applied and Computational Mathematics and Statistics University of Notre Dame, Notre Dame, Indiana 46556, USA CHUAN XUE Department of Mathematics and Mathematical Biosciences Institute Ohio State University, Columbus, OH 43210, USA ABSTRACT. This paper is concerned with a three-dimensional model of wound healing. The boundary of the wound is a free boundary, and the region surrounding it is viewed as a partially healed tissue, satisfying a viscoelastic constitutive law for the velocity v. In the partially healed region the densities of several types of cells and the concentrations of several chemical species satisfy a coupled system of parabolic equations, whereas the tissue density satisfies a hyperbolic equation. The parabolic equations include advection by the velocity v and chemotaxis/haptotaxis terms. We prove existence and uniqueness of a smooth solution of the free boundary problem, for some time interval 0 t T, T > 0. We also simulate the model equations to demonstrate the difference in the healing rate between normal wounds and chronic (or ischemic) wounds. 1. Introduction. Wound healing under normal conditions is a process consisting of four overlapping stages: haemostasis, inflammation, proliferation and remodeling [5, 13, 19]. During haemostasis, which occurs immediately after injury, clotting factors are delivered by platelets to the injured site to stop bleeding. Platelets also release chemokines, such as platelet-derived growth factor (PDGF), which recruits blood-borne cells to the wound. During the inflammatory phase, macrophages migrate into the wound, remove necrotic tissue and kill infectious pathogens. They also enhance the production of growth factors secreted by platelets, such as vascular endothelial growth factors (VEGFs), to attract fibroblasts and endothelial cells towards the wound. The proliferative phase is characterized by the production of extracellular matrix (ECM) by fibroblasts, and by the directed growth and movement of new blood vessels (angiogenesis) into the wound. The newly deposited ECM serves as a bed for tissue repair, and also contributes to scar formation. During the remodeling phase, which may take several years, fibroblasts and other cells interact to increase the tensile strength of the ECM. Chronic wounds are those that fail to proceed through the above four stages due to venous insufficiency [18, 21]. Chronic wounds represent a major 2000 Mathematics Subject Classification. Primary: 92C50, 35R35, 35B40. Key words and phrases. Wound healing, ischemia, viscoelasticity, free boundary problems, existence and uniqueness of solutions. This work is supported by NSF grant and the National Institute of Health (Office of the Director) Award UL1RR

2 2 AVNER FRIEDMAN, BEI HU AND CHUAN XUE public health problem affecting 6.5 million individuals in the United States. It is estimated that $25 billion is spent annually on the treatment of chronic wounds [18]. Oxygen perfusion depends on the formation of new blood vessels that move into the wound, and it plays a critical role in the healing process. There are several mathematical models of wound healing which incorporate the effect of angiogenesis [15, 14, 3, 17]. Mathematical models of angiogenic networks, such as through the induction of vascular networks by VEGFs [6, 7], were developed by McDougall and coworkers [12, 20, 2], in connection with chemotherapeutic strategies. The role of oxygen in wound healing was explicitly incorporated in [3] and [17]. In particular, it was demonstrated in [17] that enhanced healing can be achieved by moderate hyperoxic treatment. A recent model by Xue, Friedman and Sen [22], which builds on [17], includes also the velocity of the ECM and treats the wound boundary as a free boundary. This model was initially developed for twodimensional radially symmetric wounds, and the numerical results were shown to be in agreement with the experimental measurements for cutaneous wounds [16]. More recently the model was extended to three-dimensional wounds [11]. In the present paper we consider this three-dimensional model. We establish the existence and uniqueness of a solution for some time interval 0 t T, T > 0. We then proceed to simulate the model for both normal and chronic wounds. The simulation results focus on two variables: the decreasing region of the open wound, and the macrophage density at the free boundary, that is, at the boundary of the open wound. As in the two-dimensional case [22], the three-dimensional simulations demonstrate the difference in the healing process between normal and chronic wounds. In Section 2 we introduce the three-dimensional mathematical model developed in [11]. The model consists of two coupled systems of PDEs. The first system is a strongly elliptic system for the three components of the velocity v in the evolving partially healed region Ω t. This system involves the isotropic pressure P, which is a function of the ECM density ρ. The second system involves various cell densities and concentrations of growth factors, oxygen, and ρ. It is a parabolic system for 7 variables and one hyperbolic equation for ρ. All these equations include advection terms by the velocity v. In Sections 3-4 we consider the free boundary problem for the strongly elliptic system for the three components of the velocity v, assuming that the pressure P is a given function; we call this system the reduced problem. In Section 3 we establish some properties of this system, and in Section 4 we prove existence and uniqueness of a solution of the reduced problem for some time interval 0 t T, T > 0. In Section 5 we use the results of Section 4 to prove existence and uniqueness for the full wound healing problem, i.e., for the two coupled PDE systems. The special case of axially symmetric wounds is described in Section 6. this case is simulated in Section 7 for both normal and chronic wounds. 2. Model equations. We introduce the following variables: - w(t, x): concentration of tissue oxygen - e(t, x): concentration of Vascular Endothelial Growth Factor (VEGF) - p(t, x): concentration of Platelet-Derived Growth Factor (PDGF) - m(t, x): density of macrophages - f(t, x): density of fibroblasts - n(t, x): density of capillary tips - b(t, x): density of capillary sprouts - ρ(t, x): density of the ECM - v(t, x): velocity of the ECM

3 A THREE DIMENSIONAL MODEL OF WOUND HEALING 3 A t W t Ω t r = L z = H 0 FIGURE 1. The geometry of the wound. The geometry of the wound is shown schematically in Figure 1. The open wound, W t, is given by W t = {(x, y, z) : Z(t, x, y) < z < 0, (x, y) A t }. The wound s boundary is Γ t = W t {z < 0}, and the partially healed region is Ω t = D\W t, where D = {(x, y, z) : H 0 < z < 0, x 2 + y 2 < L} is a cylindrical domain whose boundaries x 2 + y 2 = L and z = H 0 border the normal healthy tissue. The ECM in Ω t is a growing collagen matrix which is elastic on a short time scale and viscous on a long time scale. We shall model it as upper convected Maxwell (viscoelastic) fluid with isotropic pressure depending on its density. The mass conservation law for the ECM density ρ is ρ t + (ρv) = G ρ(f, w, ρ), (2.1) where G ρ = k ρw f(1 ρ ) λ ρ ρ, (2.2) w + K wρ ρ m and k ρ, K wρ, ρ m, λ ρ are positive constants. The momentum equation is (ρv) + (ρv v) = σ, t where σ is the total stress. We can write σ = P I +τ where P is the isotropic pressure and τ is the deviatoric stress. Since healing is a slow or quasi-stationary process with negligible inertia, the last equation can be approximated by σ = 0, or σ = P + τ = 0 (2.3) For compressible material the isotropic pressure is a function of the density, i.e., P = P (ρ), and we take ( ) ρ P = βf 1, (2.4) ρ 0 where β, ρ 0 are positive constants, and 0, x < ε, x 3 F (x) = 6ε 2 + x2 2ε + x 2 + ε 6, ε x < 0, x3 6ε 2 + x2 2ε + x 2 + ε 6, 0 x < ε, x, x ε.

4 4 AVNER FRIEDMAN, BEI HU AND CHUAN XUE with ε/ρ 0 1 is a C 2+α approximation to x + for any α (0, 1). For an upper convected Maxwell fluid, the stress-strain relationship is given by ( ) Dτ λ ( v)τ τ( v)t + τ = η( v + v T ), Dt where η is the shear viscosity, and, as shown in Xue et al. [22], the first term on the left-hand side is very small, so after dropping it we obtain, Hence, (2.3) is equivalent to, Set x 1 = x, x 2 = y, x 3 = z. Then (2.6) is equivalent to η 3 i=1 x i τ = η( v + v T ). (2.5) η ( v + v T ) P = 0. (2.6) ( vj + v ) i P = 0, j = 1, 2, 3. (2.7) x i x j x j Denote the free boundary by φ(x, y, z, t) = 0. By continuity, the free boundary points move with the velocity v at these points. Hence, the equation for the free boundary is φ t + v φ = 0 on Γ t. (2.8) We assume that the external force at the wound surface is zero and there is no surface tension; hence, σ ν = 0. Hence, the boundary condition for v is, or η 3 i=1 η ( v + v T ) ν P ν = 0 on Γ t, (2.9) ( vj + v ) i ν i P ν j = 0, j = 1, 2, 3, on Γ t. (2.10) x i x j If we assume that the free boundary can be written as z = Z(t, x, y), then φ = Z(t, x, y) z, and ν = ( Z x, Z y, 1)/ Zx 2 + Zy The boundary conditions for v at the fixed boundaries are v 1 = v 2 = v 3 = 0 v 1 = v 2 = 0, v 3 = 0 on {z = H 0 } { x 2 + y 2 = L}, on {z = 0, (x, y) / A t, x 2 + y 2 < L}, (2.11) All the boundaries are characteristic curves for (2.1), thus no boundary conditions are imposed for ρ. The initial condition for ρ is ρ(0, x) = ρ 0. (2.12) In summary, the equations for ρ and v are (2.1), (2.6), (2.9), (2.11) and (2.12), while the free boundary equation is (2.8).

5 A THREE DIMENSIONAL MODEL OF WOUND HEALING 5 In addition to the equations for ECM, the remaining variables listed above satisfy the following system of partial differential equations in Ω t (see [22, 11]), w t + (wv ) = (D w w) + B w (w, f, m, p), (2.13) p t + (pv ) = (D p p) + B p (w, f, m, p), (2.14) e t + (ev ) = (D e e) + B e (w, m, e, b, n), (2.15) m t + (mv ) = (D m m mχ m p) + B m (w, m, p, b), (2.16) f t + (fv ) = (D f f fχ f p) + B f (w, f), (2.17) n t + (nv ) = (D n n nχ n e) + B n (b, n, e), (2.18) b t + (bv ) = ( D b b + Ab b 0 (D n n nχ n e) ) + B b (w, b, n),(2.19) where B w (w, f, m, p) = k w b ( w b w ) [ (λwf f + λ ) ( wmm 1 + λ ) ] wwp + λ wm w, K p + p ( ) w B p (w, f, m, p) = k p mg p λ pf fp w 0 K p + p λ pp, B e (w, m, e, b, n) = k e mg e ( w w 0 ) (λ en n + λ eb b + λ e )e, B m (w, m, p, b) = k mbp K p + p λ mm (1 + λ d D(w)), ( B f (w, f) = k f G f (w)f 1 f ) λ f f(1 + λ d D(w)), f m e B n (b, n, e) = (k nb b + k n n) K e + e (λ nbb + λ nn n)n, ( B b (w, b, n) = k b G b (w)b 1 b ) + (λ nb b + λ nn n)n, b 0 ρ ρ χ m (m, ρ) = χ m H(m m m), χ f (f, ρ) = χ f H(f m f), ρ 0 ρ 0 ρ χ n (n, ρ) = χ n H(n m n), ρ 0 the χ i s are chemotactic/haptotactic coefficients, and H(s) is the smooth approximation of the Heaviside function, u 6 H(u) = u 6, u 0 0, u < 0. The level of oxygen in the wound is a critical factor in the healing process. Moderate hypoxia improves healing; it stimulates macrophages to produce growth factors. Severe hypoxia impairs healing, since there is not enough oxygen for cells to grow and proliferate. Moderate hyperoxia improves healing, as it enables cells to proliferate faster. These facts should be incorporated by taking the profile of G e (w) as in figure 2; a similar profile should

6 6 AVNER FRIEDMAN, BEI HU AND CHUAN XUE be assumed for G p (w). For simplicity we approximate such profiles by taking, 3w, 0 w < 0.5 2w, 0 w < 0.5, 2 w, 0.5 w < 1 2 2w, 0.5 w < 1, G p (w) = 1 3 w + 2 3, 1 w < 4, G e (w) = 1 3 w 1 3, 1 w < 4,, 2, w 4 1, w 4 We also take G f (w) = (K wf + 1)w K wf + w, G b = (K wρ + 1)w, D(w) = 1 H(5w 1)H(1 w/3). K wρ + w G e w hypoxic normoxic hyperoxic FIGURE 2. The profile of G e (w). To complete the model we need to prescribe boundary and initial conditions for the above variables. On the wound s boundary Γ t, PDGF is secreted by platelets, but secretion rate diminishes as the wound closes. We take D p p ν = g( W t,z ) on Γ t, (2.20) where ν is the outward normal vector, and W t,z is the area of the cross section W t,z of the open wound with the horizontal plane at depth z and time t. The function g is a monotone decreasing function of W t,z and g(0) = 0. The remaining variables satisfy w ν = e ν = n ν = b ν = 0 on Γ t m D m ν mχ p m ν = D f f ν fχ f p ν = 0 on Γ t. The boundary conditions on the fixed boundaries for p, e, w, m, f, n, b are (2.21) w/w 0 = f/f 0 = b/b 0 = 1 on {z = H 0 } { x 2 + y 2 = L}, (2.22) p = e = m = n = 0 on {z = H 0 } { x 2 + y 2 = L}, (2.23) w ν = e ν = p ν = m ν = f ν = n ν = b ν = 0 on {z = 0, (x, y) / A t}. (2.24) where f 0, b 0 are the constant densities of f, b for normal healthy tissues. We impose the following initial conditions: f/f 0 = w/w 0 = 1, e = m = n = 0 in Ω 0, b = b 0 (x), p = p 0 (x) in Ω 0, (2.25)

7 A THREE DIMENSIONAL MODEL OF WOUND HEALING 7 where b 0 (x), p 0 (x) have three continuous derivatives and satisfy the boundary conditions (2.20) - (2.24), and b 0 (x) = b 0, p 0 (x) = 0 in Dist{x; Γ t } > ε. Ischemia is a condition where blood supply to organ or tissue is decreased as a result of constriction or obstruction of blood vessels. We associate chronic wounds with ischemia, and, accordingly, we change the source function B w and the boundary conditions on { x 2 + y 2 = L} and {z = H 0 } as follows. B w = k w b ( (1 γ)w b w ) [ (λwf f + λ ) ( wmm 1 + λ ) ] wwp + λ wm w, (2.26) K p + p (1 γ 1 )(w w 0 ) + γ 1 L w ν = 0 on x 2 + y 2 = L, (1 γ 2 )(w w 0 ) + γ 2 H 0 w ν = 0 on z = H 0, (2.27) where ν is the unit outward normal on the boundary, γ 1, γ 2 [0, 1] are the ischemic parameters, and γ is a linear combination of γ 1 and γ 2, i.e., γ = aγ 1 +(1 a)γ 2, for some constant 0 < a < 1. Similar boundary conditions are imposed for the variables p, e, m, f, n, b. Notice that if γ i = 1 for i = 1, 2, then the fluxes across { x 2 + y 2 = L} and {z = H 0 } are zero, which means a total cutoff of blood supply. Moderate ischemia is represented by an intermediate value of γ i. The above ischemic boundary conditions can be formally derived by homogenization [22, 9]. Definition 2.1. We shall refer to the system of (2.1), (2.6), (2.8), (2.13) - (2.19), with the boundary conditions (2.9), (2.11),(2.20) - (2.24), and initial condition (2.12), (2.25) as the normal wound model. The corresponding model when B w is given by (2.26) and the boundary conditions on the fixed boundaries are replaced by (2.27) for w and similar Robin conditions for p, e, m, f, n, b will be called the ischemic wound model. In the following three sections we prove the existence and uniqueness of the solution for the normal wound model for a small time interval 0 t T, T > 0. The proof for the ischemic wound model is similar. 3. The reduced problem and its properties. We first consider the subsystem (2.6), (2.9), and (2.11) (or, equivalently, (2.7), (2.10), and (2.11)) with the free boundary condition (2.8), and prove local existence and uniqueness. Later on we shall extend the proof to the complete normal wound model. Without loss of generality, we assume from now on that η = 1. Definition 3.1. The system (2.6) with the boundary conditions (2.8), (2.9), (2.11), and the initial domain Ω 0 will be called the reduced problem. Definition 3.2. The system (2.6) in Ω t with the boundary conditions (2.9) and (2.11) when t is a fixed time will be called the static reduced problem Properties of the static reduced problem. We first consider the static reduced problem and show that the system is strongly elliptic, satisfies the Supplementary Condition (see [1], p.39), the Complementing Boundary Condition [1, p.42], and that the only solution of the homogeneous system is zero. This will ensure (by [1]) that for any smooth boundary z = Z(t, x, y) and a smooth function P (t, x) (t fixed) the static reduced problem has a unique smooth solution.

8 8 AVNER FRIEDMAN, BEI HU AND CHUAN XUE Strong ellipticity. The matrix l corresponding to the principal part of the operators in the equations in (2.7) is l (Ξ) = 2ξ2 1 + ξ2 2 + ξ3 2 ξ 1 ξ 2 ξ 1 ξ 3 ξ 1 ξ 2 ξ ξ2 2 + ξ3 2 ξ 2 ξ 3, (3.1) ξ 1 ξ 3 ξ 2 ξ 3 ξ1 2 + ξ ξ3 2 where Ξ = (ξ 1, ξ 2, ξ 3 ), and its determinant is L(Ξ) = det(l (Ξ)) = 2(ξ ξ ξ 2 3) 3 = 2 Ξ 6. (3.2) The system (2.7) is strongly elliptic since l (Ξ) has three positive eigenvalues λ 1 = 2 Ξ 2, and λ 2 = λ 3 = Ξ 2 for all Ξ 0. Supplementary condition. For any linearly independent vectors Ξ = (ξ 1, ξ 2, ξ 3 ), Ξ = (ξ 1, ξ 2, ξ 3), and τ C, we construct the polynomial L(Ξ + τξ ) in τ for the system (2.7), ( L(Ξ + τξ ) = 2 τ 2 Ξ 2 + 2(Ξ Ξ )τ + Ξ 2) 3. The system (2.7) satisfies the Supplementary Condition [1, p.39] if and only if the polynomial L(Ξ + τξ ) in the complex variable τ has exactly three roots with positive imaginary part. But since Ξ Ξ Ξ Ξ and equality holds if and only if Ξ and Ξ are linearly dependent, 4(Ξ Ξ ) 2 4 Ξ 2 Ξ 2 < 0. Hence, the polynomial L(Ξ+τΞ ) in τ has three roots with positive imaginary part, namely, τ + 1 = τ + 2 = τ + 3 = Ξ Ξ + i (Ξ Ξ ) 2 + Ξ 2 Ξ 2 Ξ 2. (3.3) Complementing boundary condition. In the following we show that the boundary conditions (2.9) on the free boundary Γ t satisfy the Complementing Boundary Condition [1, p.42]. One can similarly, and, in fact, more easily show that the boundary conditions in (2.11) also satisfy the Complementing Boundary Condition. We begin with some computations. The (inward) normal vector to the free boundary is and the tangential vectors are m = n = 1 (Z x, Z y, 1), 1 + Zx 2 + Zy 2 1 µ µ (µ 1Z x + µ 2 Z y ) 2 (µ 1, µ 2, µ 1 Z x + µ 2 Z y ), for any (µ 1, µ 2 ) (0, 0). Since m n = 0, by (3.3) the roots τ + k As in [1, p.42], set M + = τ + 1 = τ + 2 = τ + 3 = i. of L(m + τn) = 0 are 3 (τ τ + j ) = (τ i)3 = τ 3 3i τ 2 3τ + i. (3.4) j=1

9 A THREE DIMENSIONAL MODEL OF WOUND HEALING 9 Then, mod M +, is τ 3 = 3i τ 2 + 3τ i, (3.5) τ 4 = 6τ 2 + 8i τ + 3, (3.6) τ 5 = 10i τ 2 15τ + 6i. (3.7) The matrix B (Ξ) corresponding to the principal part of the boundary operators in (2.10) B (Ξ) = 2ξ 1 Z x + ξ 2 Z y ξ 3 ξ 1 Z y ξ 1 ξ 2 Z x ξ 1 Z x + 2ξ 2 Z y ξ 3 ξ 2 ξ 3 Z x ξ 3 Z y ξ 1 Z x + ξ 2 Z y 2ξ 3 ξ 1 = (ξ 1 Z x + ξ 2 Z y ξ 3 )I 3 + ξ 2 (Z x, Z y, 1), (3.8) ξ 3 where I 3 is the 3 3 unit matrix. The adjoint matrix to l (Ξ) is ξ 2 adj l (Ξ) = (ξ1 2 + ξ2 2 + ξ3) ξ ξ3 2 ξ 1 ξ 2 ξ 1 ξ 3 ξ 1 ξ 2 2ξ1 2 + ξ ξ3 2 ξ 2 ξ 3 ξ 1 ξ 3 ξ 2 ξ 3 2ξ ξ2 2 + ξ3 2 = Ξ 2 2 Ξ 2 I 3 ξ 1 ξ 2 ξ 3 (ξ 1, ξ 2, ξ 3 ), (3.9) where Ξ = (ξ 1, ξ 2, ξ 3 ) as before. Then for Ξ = m+τn, we have m n = 0, (Z x, Z y, 1) = 1 + Z 2 x + Z 2 y n, and Thus where Ξ 2 = m 2 + τ 2 n 2 = 1 + τ 2, (3.10) Ξ (Z x, Z y, 1) = 1 + Zx 2 + Zy 2 Ξ n = τ 1 + Zx 2 + Zy 2. (3.11) B (Ξ)adj l (Ξ) = 2 Ξ 2 Ξ (Z x, Z y, 1) Ξ 2 I 3 ξ 2 (ξ 1, ξ 2, ξ 3 ) ξ 3 +2 Ξ 4 ξ 1 ξ 2 ξ 3 (Z x, Z y, 1) = Zx 2 + Zy(1 2 + τ 2 ) { τ(1 + τ 2 )I 3 } τ(m + τn) T (m + τn) + (1 + τ 2 )(m + τn) T n = Zx 2 + Zy 2 ξ 1 5 τ j Q j, (3.12) j=0 Q 5 = I 3, Q 4 = n T m, Q 3 = 2I 3 m T m + n T n, Q 2 = n T m + m T n, Q 1 = I 3 m T m + n T n, Q 0 = m T n.

10 10 AVNER FRIEDMAN, BEI HU AND CHUAN XUE Substituting (3.5) (3.7) into (3.12) we obtain where B (Ξ)adj l (Ξ) = Zx 2 + Zy 2 2 τ j Qj, mod M +. j=0 Q 2 = 3iQ 3 6Q 4 10iQ 5 + Q 2, Q 1 = 3Q 4 + 8iQ 4 15Q 5 + Q 1, Q 0 = iq 3 + 3Q 4 + 6iQ 5 + Q 0. The system is said to satisfy the complimentary boundary condition if the rows of the matrix B (Ξ)adj l (Ξ) are linearly independent mod M + (τ); that is, if 3 C k k=1 j=0 2 τ j Qk j 0 mod M + (τ), where Q k j is the k-th row vector of the matrix Q j, then all the constants C k are zero. Suppose not all the C k are zero. Then 2 det τ j Qj 0, j=0 τ, and, in particular, det Q 0 0. However, Q 0 = 4iI 3 + im T m in T n 3n T m + m T n. Writing n = (n 1, n 2, n 3 ), m = (m 1, m 2, m 3 ) and noticing n n n 2 3 = 1, m m m 2 3 = 1 and n 1 m 1 + n 2 m 2 + n 3 m 3 = 0, we find that det Q 0 = 48i 0, (3.13) which is a contradiction. This verifies the supplementary boundary condition. The homogeneous problem. We claim that the homogeneous system for v, 3 i=1 x i with the homogeneous boundary conditions 3 i=1 ( vj + v ) i = 0, in Ω t, j = 1, 2, 3. (3.14) x i x j ( vj + v ) i ν i = 0, on Γ t, j = 1, 2, 3, (3.15) x i x j v 1 = v 2 = v 3 = 0, on {z = H 0 } { x 2 + y 2 = L}, (3.16) v 1 = v 2 = 0, v 3 = 0, on {z = 0, (x, y) / A t, x 2 + y 2 < L}, (3.17)

11 A THREE DIMENSIONAL MODEL OF WOUND HEALING 11 has only the trivial solution v = 0. To prove it, we multiply (3.14) by v j, sum over index j, and integrate over the domain Ω t. We obtain 0 = = 3 Ω t i,j=1 3 v j Ω t i,j=1 x i ( vj + v ) i dx x i x j ( vj v j ν i + v i x i x j Ω t ) ds (3.18) [ 3 ( vj ) 2 ( ) ( ) ] vj vi + dx, x i x i x j i,j=1 where ν = (ν 1, ν 2, ν 3 ) is the outward normal at the boundary of the domain Ω t. On the boundary {z = H 0 } { x 2 + y 2 = L}, we have v j = 0, by (3.16); on the boundary {z = 0, (x, y) A t, x 2 + y 2 < L}, we have ν = (ν 1, ν 2, ν 3 ) = (0, 0, 1), and v 1 = v 2 = v 3 = v 3 x = v 3 = 0, by (3.17); finally on the boundary z = Z(t, x, y), ( y 3 vj i=1 + v ) i ν i = 0, by (3.15). Thus, all the boundary integrals in (3.18) vanish, x i x j and (3.18) reduces to [ 3 ( vj ) 2 ( ) ( ) ] vj vi + dx = 0, (3.19) Ω t i,j=1 x i x i x j Since [ 3 ( vj ) 2 ( ) ( ) ] vj vi + = i,j=1 x i 3 ( ) 2 vi 2 + x i i=1 x i 3 i,j=1,i j x j ( 1 vj + v ) 2 i, 2 x i x j we conclude v i 0 and v j + v i 0 if i j. By the boundary conditions (2.11), x i x i x j we have v 1 (y, z) = 0, for y = ± L 2 x 2, H 0 < z < 0, 0 < x < L, hence v 1 0. Similarly v 2 0, and then v 3 x = v 1 = 0, v 3 y = v 2 = 0, so that v 3 const. = 0. This completes the proof that the homogeneous system has only the trivial solution v 1 = v 2 = v Existence and uniqueness of solution for the (extended) reduced system. It is useful to extend the reduced problem to Ω t = { H 0 < z < H 0, r < L}\{Z(x, y, t) z

12 12 AVNER FRIEDMAN, BEI HU AND CHUAN XUE Z(x, y, t)} by defining { vj (t, x, y, z) if z < 0, V j (t, x, y, z) = v j (t, x, y, z) if z > 0, j = 1, 2, { v3 (t, x, y, z) if z < 0 V 3 (t, x, y, z) = v 3 (t, x, y, z) if z > 0, P (t, x, y, z) = P (t, x, y, z) if z > 0, { Z(t, x, y) z if z 0 Φ(t, x, y, z) = Z(t, x, y) + z if z > 0, Then V = (V 1, V 2, V 3 ) and Φ satisfy the system (4.1) ( V + V T ) P = 0 in Ω t, (4.2) ( V + V T ) ν P ν = 0 on Φ(t, x, y, z) = 0, (4.3) V 1 = V 2 = V 3 = 0 on {z = ±H 0 } { x 2 + y 2 = L}, (4.4) Φ t + V Φ = 0, on the free boundary Γ t = {Φ = 0}. (4.5) Definition 4.1. We shall call the system (4.2) - (4.5) the extended reduced problem. Lemma 4.2. If the extended reduced problem has a unique solution (in some differentiability class), then the reduced problem has a unique solution (in the same differentiability class). Proof. Assume V = (V 1, V 2, V 3 ) and Φ form the unique solution of (4.2) - (4.5). Let Ṽ 1, Ṽ2, Φ be the reflections of V 1, V 2, Φ, and Ṽ3 be the anti-reflections of V 3 across z = 0. Then Ṽ = (Ṽ1, Ṽ2, Ṽ3), Φ also form a solution of (4.2) - (4.5). By uniqueness, Ṽ = V, Φ = Φ. Hence v = V z 0 satisfies the boundary conditions (2.11) at z = 0, and thus v and φ = Φ z 0 form a solution of the reduced problem. The solution of the reduced problem is unique, since two different solutions of the reduced problem can be extended (by (4.1)) to two different solutions of the extended reduced problem. Remark 1. Lemma 4.2 does not actually depend on the fact that the free boundary is of the form z = Z(t, x, y); the reflection of the free boundary across z = 0 can be done with any parametric representation. In the next section we shall, more conveniently, represent the free boundary in the form x = X(t, θ, φ) Existence and uniqueness of solution for the extended reduced problem. In this section, we prove that the extended reduced problem has a unique solution, for a small time interval, with the free boundary parametrized as x = X(t, λ), where λ = (θ, φ) Λ, Λ = { π/2 θ π/2, 0 φ 2π}. We assume that the initial free boundary X 0 (λ) = X(0, θ, φ)}, π/2 θ π/2, 0 φ 2π is orthogonal to the (x, y) plane at x = X(0, 0, φ). (4.6) Then the extension of the domain by reflection across z = 0 does not have corners at the points x = X(0, 0, φ). We further assume that for some α (0, 1), and X(0, ) is in C 3+α (Λ), X(0, λ) const > 0, λ Λ. (4.7) P 1 sup P (t, ) C 1+α (D ) <, (4.8) 0 t T 0

13 A THREE DIMENSIONAL MODEL OF WOUND HEALING 13 where D = {(x, y, z), 0 x 2 + y 2 < L, H 0 < z < H 0 }. We introduce the local orthogonal unit vectors e r (λ), e θ (λ), e φ (λ) in the directions of increasing r, θ, φ, respectively. We write surfaces X(t, λ), λ Λ in a neighborhood of X(0, λ) in the form Then and X(t, λ) = X 0 (λ) + h(t, λ)e r (λ). (4.9) X t (t, λ) = h t (t, λ)e r (λ), or h t (t, λ) = X t (t, λ) e r (λ), (4.10) X θ (t, λ) = X 0 θ(λ) + h θ (t, λ)e r (λ) + h(t, λ)e θ (λ), (4.11) X φ (t, λ) = X 0 φ(λ) + h φ (t, λ)e r (λ) + h(t, λ)(cos(θ))e φ (λ), (4.12) so that, by taking scalar product with e r, h θ (t, λ) = X θ (t, λ) e r (λ) X 0 θ(λ) e r (λ), (4.13) h φ (t, λ) = X φ (t, λ) e r (λ) X 0 φ(λ) e r (λ), (4.14) It will be convenient to express the free boundary condition (4.5) as a dynamic equation for h. To do that, suppose a point X(t, λ ) moves to position X(t + t, λ) after a short time t, where λ = (θ, φ), λ = (θ, φ ), and introduce the point λ = (θ, φ). Writing we obtain X(t + t, λ) X(t, λ) t = X(t + t, λ) X(t, λ ) t X(t, λ) X(t, λ ) t X t (t, λ) = X(t + t, λ) X(t, λ) lim t 0 t = X(t + t, λ) X(t, λ ) lim t 0 t lim t 0 X(t, λ) X(t, λ ) t lim t 0 X(t, λ ) X(t, λ ), t X(t, λ ) X(t, λ ). (4.15) t By the continuity condition (which states that the free boundary at X(t, λ) moves with the velocity V at X(t, λ)), and X(t, λ) X(t, λ ) lim t 0 t X(t, λ ) X(t, λ ) lim t 0 t X(t + t, λ) X(t, λ ) lim = V(t, X(t, λ)), (4.16) t 0 t = ( lim t 0 X(t, λ) X(t, λ ) ( ) θ θ lim t 0 = X θ(t, λ) (V(t, X(t, λ)) e θ ), X(t, λ) ( X(t, λ ) X(t, λ ) ( ) = lim t 0 φ φ = X φ(t, λ) (V(t, X(t, λ)) e φ ), X(t, λ) cos θ lim t 0 θ θ ) t φ φ ) t (4.17) (4.18)

14 14 AVNER FRIEDMAN, BEI HU AND CHUAN XUE Hence X t (t, λ) = V(t, X(t, λ)) X θ(t, λ) (V(t, X(t, λ)) e θ ) X(t, λ) X φ(t, λ) (V(t, X(t, λ)) e φ ). X(t, λ) cos θ Taking the scalar product with e r and using (4.10), (4.13) and (4.14), we obtain (4.19) h t (t, λ) + V(t, X(t, λ)) e θ h θ (t, λ) + V(t, X(t, λ)) e φ X(t, λ) X(t, λ) cos θ h φ(t, λ) = V(t, X(t, λ)) e r (4.20) ( )( ) ( )( ) X 0 θ (λ) e r V(t, X(t, λ)) eθ X 0 φ (λ) e r V(t, X(t, λ)) eφ X(t, λ) X(t, λ) cos θ Conversely, one can deduce from (4.19), or (4.20), the free boundary condition written in the form V ν = V ν, where V ν is the velocity of the free boundary in the normal direction ν. Indeed, by (4.15) - (4.18) we have X(t + t, λ) X(t, λ ) V ν = lim ν t 0 [ t X(t, λ) X(t, λ ) = X t (t, λ) + lim t 0 t = + lim t 0 X(t, λ ) X(t, λ ] ) ν t [ X t (t, λ) + X θ(t, λ) (V(t, X(t, λ)) e θ ) + X φ(t, λ) (V(t, X(t, λ)) e φ ) X(t, λ) X(t, λ) cos θ = V(t, X(λ, t)) ν. ] ν Given any family of curves X(t, λ) as in (4.9), we denote by Ω t the domain bounded by X(t, λ) and the boundary of the rectangle D, and write the extended system (4.1) - (4.4) in the form of where f 1, g 1 are linear functions. We introduce a class of functions h(t, λ) by where Q t V = f 1 ( P ) in G(t) in Ω t, (4.21) B t V = g 1 (P ) on G(t) on Ω t, (4.22) W T,M1,M = {h(t, λ); h(0, λ) = 0, sup h(t, ) C 2+α (Λ) M 1, h T M}, 0 t T h T = sup h(t, ) C 2+α (Λ) + sup h t (t, ) C 1+α (Λ), M, M 1 > 0,. 0 t T 0 t T For any h W T,M1,M, consider the problem (4.21), (4.22). From the results of Section 3 we know that for each t, the system (4.21), (4.22) is a strongly elliptic systems satisfying the Supplementary Condition and the complimenting boundary condition. Hence we can apply the Schauder estimates [1] to obtain the following lemma. Lemma 4.3. Under the conditions (4.6) - (4.8), there exists a unique solution V(t, x) of (4.21), (4.22) for 0 t T, such that V(t, ) C 2+α (Ω t ) C(M 1 ) P (t, ) C 1+α (D ) (4.23) where C(M 1 ) depends on M 1 = sup 0 t T h(t, ) C 2+α (Λ)..

15 A THREE DIMENSIONAL MODEL OF WOUND HEALING 15 We next extend the function V(t, x) into D such that V(t, ) C 2+α (D ) c 0 C(M 1 )P 1, c 0 > 0. (4.24) For simplicity, we denotes the extended function also by V. We shall need to use the same extension procedure for any V corresponding to any h in W T,M1,M. We can achieve such an extension by first extending V along each normal to the curve X(t, λ) by a polynomial of degree two in the distance along this normal (to achieve smooth C 2+α extension) and then multiply this extension by a fixed cutoff C 3 function which equals to 1 in Ω t and vanishes outside a small neighborhood of Ω t, independent of t, 0 t T (T small). We define a function h(t, λ) by h t (t, λ) + V(t, X(t, λ)) e θ h θ (t, λ) + V(t, X(t, λ)) e φ X(t, λ) X(t, λ) cos θ h φ (t, λ) ( )( ) X 0 = V(t, X(t, λ)) e r θ (λ) e r V(t, X(t, λ)) eθ X(t, λ) ( )( ) (4.25) X 0 φ (λ) e r V(t, X(t, λ)) eφ h(0, λ) = 0, X(t, λ) cos θ where V is the extension defined above. We note that the assumption (4.7) will be needed in order to prove that h(t, λ) is in C 2+α. We introduce the mapping S : h h. Clearly h is a fixed point of S if and only if the corresponding V and X(t, λ) form a solution of the extended reduced free boundary problem. To prove that S has a fixed point, we use the following lemma. Lemma 4.4. Consider the hyperbolic equation with initial condition (i) Assume that w t + b(t, λ) λ w = G(t, λ), λ Λ, 0 t T, T > 0 (4.26) w t=0 = 0. b(t, ) C 2+α (Λ) + G(t, ) C 2+α (Λ) K, t [0, T ]. Then there exists a unique solution of (4.26) satisfying where C 1 (K) depends only on K. (ii) If, in addition, then w(t, ) C 2+α (Λ) C 1 (K)T, (4.27) b(t, ) C 2+α,α/2 (Λ,t)(Λ [0,T ]) + G(t, ) C 2+α,α/2 (Λ,t)(Λ [0,T ]) K, where C 2 (K) depends only on K. w C 2+α,α/2 (Λ,t)(Λ [0,T ]) C 2 (K)T 1 α/2, (4.28) Proof. The proof of (i) is obtained by applying the proof of Lemma 2.2 in [4], or Lemma 3.2 in [8]. The only part of the lemma which requires an additional argument is the proof of (4.28). To prove this estimate we set W (t, λ) = D 2 λ[w(t + t, λ) w(t, λ)],.

16 16 AVNER FRIEDMAN, BEI HU AND CHUAN XUE and proceed to estimate it in the same way as W (t, λ) = D 2 λ[w(t, λ + λ) w(t, λ)]. in the proof of (i). Since the characteristic curves through (t, λ) and (t+ t, λ) are as close to each other as the characteristic curves through (t, λ) and (t, λ + λ) (up to a multiplicative constant), the only difference in estimating W and W is the fact that W (0, λ) 0 whereas W (0, λ) 0. This however causes no problem since, by integrating the derivative of W (s, λ) along the characteristic curve, which passes through ( t, λ), we find that W (0, λ) const λ const λ α/2 T 1 α/2. Applying Lemma 4.4 to h we get h(t, ) C 2+α (Λ) C 1 (M 1 )T, and then, from (4.25) we then also get the bound h t (t, ) C 1+α (Λ) C 2 (M 1 ); here the C i (M 1 ) are constants depending only on M 1. Taking T and M such that C 1 (M 1 )T < M 1, M = M 1 + C 2 (M 1 ). we conclude that S maps W T,M1,M into itself. We next show that S is a contraction, and hence, it has a unique fixed point in W T,M1,M. Take h 1, h 2 W T,M1,M and the corresponding functions X 1, V 1, h 1, Ω 1,t and X 2, V 2, h 2, Ω 2,t, and set δ = h 1 h 2 T. Since the equations are not in the same domain, it is difficult to make comparisons. Hence we shall make a change of variable to reduce the problems to the same domain, Ω 1,t by the transformation (r, θ, φ) ( r, θ, φ), where θ = θ, φ = φ, r = r (h2 h 1 )ψ(r, θ, φ), where ψ is a C function such that ψ = 1 in a neighborhood of the initial wound boundary (free boundary) and ψ = 0 near the external boundary (fixed boundary) of the domain. Clearly, this transformation pulls the domain Ω 2,t to domain at Ω 1,t, and, the regularity of this transformation is the same as the regularity of the function h 1 h 2. Furthermore, since our domain excludes the origin, this transformation will not produce any singularities for the transformed PDE system. We shall denote by Ṽ2 the function V 2 under the above change of variables. On the domain Ω 1,t, the boundary conditions of Ṽ2 under the above change of variable are the same as V 1 with error ε where ε C 1+α (Λ) C(M, P 1 ) (h 1 h 2 )(t, ) C 2+α (Λ) C(M, P 1 )δ. The quantity C(M, P 1 ) will henceforth be used to denote any constant which depends only on M and P 1. Ṽ2 satisfies the same elliptic system as V 1 except for an error term (which we conveniently view as an inhomogeneous term) whose C α norm is bounded by C(M, P 1 )δ. Hence, by Lemma 4.3, V 1 (,, t) Ṽ2(,, t) C 2+α (Ω 1,t ) C(M, P 1 )δ. Using this estimate, we can then extend the definition of V 1 and V 2 into Ω 1,t Ω 2,t such that V 1 (,, t) V 2 (,, t) C 2+α (Ω 1,t Ω 2,t ) C(M, P 1 )δ.

17 A THREE DIMENSIONAL MODEL OF WOUND HEALING 17 We now write h 1 and h 2 in integrated form along the characteristics and begin to estimate h 1 h 2, λ ( h 1 h 2 ), 2 λ ( h 1 h 2 ), and its Holder coefficient, for fixed t. We obtain, by calculation similar to [4, 8], the estimate h 1 (t, ) h 2 (t, ) C 2+α (Λ) C(M, P 1 )δt < δ 2, (4.29) if T is small enough. We have thus proved the following theorem. Theorem 4.5. Under the conditions (4.6) - (4.8), there exists a unique solution of the (extended) reduced problem for some time interval 0 t T (T > 0) with free boundary (4.9) satisfying h(t, ) C 2+α (Λ) + h t (t, ) C 1+α (Λ) M (4.30) for all 0 t T. If we replace the norm h T in W T,M by the following norm h T = h 2+α,α/2 C λ,t (Λ [0,T ]) + h t 1+α,α/2 C λ,t (Λ [0,T ]), and assume that P 2 = P 1+α,α/2 C <, (4.31) x,t (D [0,T ]) then we can repeat the proof of Theorem 4.5, and conclude that there exists a unique solution with norm h T M. In this proof we need to use the estimate (4.28) in order to estimate the (α/2)-hölder coefficient of D 2 V in t, i.e., D xix j V(x, t ) D xix j V(x, t) C t t α/2. To do that, we change the domain Ω t for V(x, t ) to Ω t for V(x, t), in the same fashion as before, and use the C α/2 Hölder condition on Dλ 2 h with respect to t. Going back to the original V(t, ) V(t, ) we then obtain Hölder estimates with respect to both x and t. We summarize this by the following theorem. Theorem 4.6. Under the conditions (4.6), (4.7), and (4.31), there exists a unique solution of the (extended) reduced problem for some time interval 0 t T (T > 0) with free boundary (4.9) satisfying h C 2+α,α/2 λ,t Furthermore, the solution V(t, x) satisfies (Λ [0,T ]) + h t 1+α,α/2 C (Λ [0,T ]) M λ,t V 2+α,α/2 C const. <, x,t (D [0,T ]) where the constant depends only on the initial surface X 0 (λ) and P Existence and uniqueness for the general case. We first extend the functions ρ, w, p, e, m, f, n, b by reflection across z = 0, and denote the extended functions by the same symbols. Because of the boundary conditions for w, p, e, m, f, n, b at z = 0, the extended functions will not have any singularities at z = 0. As in the case of the reduced problem, we solve the extended normal wound problem. Take ρ in the space Y T,M = {ρ; ρ, ρ x α,α/2 C x,t (D [0,T ]) M}. For any ρ Y T,M define P by (2.4) and use Theorem 4.6 to solve the extended reduced problem, with X(t, λ), h(t, λ), V(t, x). Then h(, λ) 2+α,α/2 C λ,t (Λ [0,T ]) + h t(, λ) 1+α,α/2 C C(M). (5.1) λ,t (Λ [0,T ])

18 18 AVNER FRIEDMAN, BEI HU AND CHUAN XUE We next extend V to all of the rectangle D = { x 2 + y 2 < L, H 0 < z < H 0 } as in the proof of Lemma 4.3. Next we solve the parabolic system for Y = (w, p, e, m, f, n, b) with given ρ, X(t, λ) (or h(t, λ)), V(t, x). Using parabolic estimates as in [10], we derive the estimate 2 j=0 DxY j α,α/2 C C(M) (t < T ) (5.2) x,t (D [0,T ]) We extend Y into D as we did for V in the proof of Lemma 4.2. We define ρ as the solution of with initial data (2.12). By Lemma 4.4 we obtain ρ t + ( ρv) = G ρ(f, w, ρ) (5.3) ρ, ρ x α,α/2 C x,t (D [0,T ]) C(M)T 1 α/2. (5.4) We extend ρ into the rectangle D in the same way we extended V in the proof of Lemma 4.3, so that the estimate (5.4) holds. Hence the mapping ρ W ρ maps Y T,M into itself if T is small. We next show that W is a contraction. We solve the parabolic system for Y 1, Y 2 corresponding to (ρ 1, h 1, V 1 ) and (ρ 2, h 2, V 2 ). One can then show that if ρ 1, ρ 2 belong to Y T,M then the corresponding V 1, V 2 and h 1, h 2 satisfy the estimates where 2 j=0 Dx(V j 1 V 2 ) α,α/2 C C(M)δ, (5.5) x,t (D [0,T ]) 1 D j t (h 1 h 2 ) C 1+α,α/2 (Λ [0,T ]) C(M)δ, (5.6) j=0 δ = ρ 1 ρ 2, D x (ρ 1 ρ 2 ) C α,α/2 x,t (D [0,T ]). To do that we use (5.5),(5.6) to estimate the difference between Y 1 and Y 2 as defined above (using also the zero initial data for Y 1 Y 2 ): 2 j=0 D j x(y 1 Y 2 ) C α,α/2 x,t (D [0,T ]) C(M)T 1 α/2 δ. (5.7) Using (5.4) - (5.7), we can estimate the difference ρ 1 ρ 2 from the equation (5.3) which they both satisfy, ρ 1 ρ 2, D x ( ρ 1 ρ 2 ) C α,α/2 x,t (D [0,T ]) C(M)T 1 α/2 δ. Hence if T is small enough then W is a contraction. The fixed point of W provides the unique solution of the extended normal wound problem. We have thus proved the following theorem. Theorem 5.1. Under the condition (4.6) - (4.7) there exists a unique solution for the normal wound problem with the free boundary satisfying the estimate (4.30) for some M > 0.

19 A THREE DIMENSIONAL MODEL OF WOUND HEALING The axially symmetric case. We claim that the PDE system is invariant under rotation about the z axis. To prove that, let A be a constant orthonormal matrix, i.e., A A T = I 3. We regard v and as row vectors. Then, written in matrix from, a rotation corresponds to a change of variables x = xa and the corresponding change of the derivative operator and velocity field are given by = A, v = va. Multiplying these equations on the right by A T, we obtain xa T = x, A T =, va T = v. (6.1) Under this change of variables, it is clear that left hand-side of (2.1) is changed to ρ t +div x(ρv) = ρ ρ + (ρv) = t t + A T (ρ va T ) = ρ t + (ρ v) = ρ t +div x(ρ v). Thus (2.1) is invariant under rotation by the matrix A. Similarly, using (6.1), and recalling the assumption that η = 1, we find that (2.6) becomes ( v + v T ) P = 0, and similarly equations (2.13)-(2.19) remain invariant under rotation above the z-axis. Hence, by the uniqueness established in Theorem 5.1, if the initial data are invariant under rotation, then the solution is also invariant under rotation. Writing the notation ρ = ρ(t, r, z), v = v 1 (t, r, z)e r + v 2 (t, r, z)e z, where r = x2 + y 2, Equation (2.1) in cylindrical coordinates becomes ρ t + 1 ( ) ( ) rρv1 + ρv2 = Gρ (f, w, ρ). (6.2) r r Writing the stress tensor τ and v in cylindrical coordinates v 1 v 2 τ rr τ rθ τ rz 0 r r τ = τ θr τ θθ τ θz, and v = v τ zr τ zθ τ zz r v 1 v, 2 0 Equation (2.5) becomes τ rr τ rθ τ rz τ θr τ θθ τ θz = τ zr τ zθ τ zz η 2η v 1 r 0 2η v 1 ( r v2 r + v ) 1 Therefore τ θr = τ rθ = τ θz = τ zθ = 0. Equation (2.3) in cylindrical coordinates reduces to ( v2 0 η 0 2η v 2 r + v ) r r (rτ rr) + (τ zr) τ θθ r P = 0, r (6.3) 1 r r (rτ rz) + (τ zz) P = 0. (6.4) Substituting the preceding expressions of τ rr, τ rz, τ θθ, τ zr, τ zz into (6.3) - (6.4), we obtain 2η 1 ( r v ) 1 + η ( v2 r r r r + v ) 1 2η v 1 r 2 P = 0, r (6.5) η 1 r r ( r v 2 r + r v 1 ) + 2η 2 v 2 2 P = 0. (6.6)

20 20 AVNER FRIEDMAN, BEI HU AND CHUAN XUE We denote the free boundary Γ t by z = Z(t, r) or by φ(t, r, z) = 0, where φ(t, r, z) = Z(t, r) z. Since the velocity of the boundary is the same as v, the equation for the free boundary is φ t + v φ = 0 on Γ t, (6.7) or Z t + v 1 Z r v 2 = 0. (6.8) The boundary conditions for v on the free boundary is derived from the relation σ ν = 0, where ν = ( Z r, 0, 1)/ Zr is the outward normal vector (pointing into W t ). In the axially symmetric case, σ is given by P + 2η v ( 1 v2 0 η r r + v ) 1 σ = σ rr σ rθ σ rz σ θr σ θθ σ θz = σ zr σ zθ σ zz η 0 P + 2η v 1 ) r ( v2 r + v P + 2η v 2 Hence the condition σ ν = 0 takes the form ( P + 2η v ) ( 1 v2 Z r η r r + v ) 1 = 0 on z = Z(r, t), (6.9) ( v2 η r + v ) 1 Z r + P 2η v 2 = 0 on z = Z(r, t). (6.10) The boundary conditions for v on the fixed boundaries are v 1 = v 2 = 0 on {z = H 0 } {r = L}, (6.11) v 1 = 0, v 2 = 0 on {z = 0, R(t, 0) < r < L}, (6.12) v 2 v 1 = 0, = 0 on {r = 0}, (6.13) r if r = R(t, z) denotes the free boundary. Equation (6.12) implies that the shear stress τ rz = 0 for z = Simulations of the model. In this section, we show simulation results of normal and ischemic wounds in the axially symmetric case, taking the coefficients in the PDE system the same as in [22]. We solved numerically the problem by the method of Arbitrary- Lagrangian-Eulerian (ALE) using COMSOL3.5a. In order to reduce the numerical error near the boundary z = 0, we solved the problem on the extended domain Ω t. Figure 3 gives the shape and macrophage density of a normal wound (γ 1 = γ 2 = 0) at day 1, 5, and 10 solved from the model. The initial wound is given by {(r, z) : r 2 + z 2 < 0.9 mm}. Figure 4 gives the shape and macrophage density of a ischemic wound with γ 1 = γ 2 = 0.6, also at day 1, 5, and 10. From these figures we see that the normal wound closes on a reasonable time scale, and macrophage goes away after day 5. However, the ischemic wound does not shrink significantly, and macrophages near the wound boundary persist over the full time course of simulation. REFERENCES [1] (MR ) S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Communications on Pure and Applied Mathematics, 17 (1964), [2] A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol., 60 (1998),

21 A THREE DIMENSIONAL MODEL OF WOUND HEALING 21 F IGURE 3. Normal wound healing. The boundary flux function of PDGF is given as g(z) = kpb R(t, z) with R0 = 0.9 mm. The boundary of the white region is given by r = R(t, z), H0 < z < H0, which is the wound boundary and its reflection across z = 0. The black curve indicates the initial position of the moving boundary, which is r2 + z 2 = R0. The color of each plot gives the macrophage density. L = H0 = 1.8 mm. F IGURE 4. Ischemic wound healing. γ1 = γ2 = 0.6. Other parameters are the same as in Figure 3. [3] H. M. Byrne, M. A. J. Chaplain, D. L. Evans and I. Hopkinson, Mathematical modelling of angiogenesis in wound healing: Comparison of theory and experiment, J. Theor. Med., 2 (2000), [4] (MR ) X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM Journal on Mathematical Analysis, 35 (2003), [5] R. F. Diegelmann and M. C. Evans, Wound healing: An overview of acute, fibrotic and delayed healing, Front. Biosci., 9 (2004), [6] Y. Dor, V. Djonov and E. Keshet, Induction of vascular networks in adult organs: Implications to proangiogenic therapy, Annals of the New York Academy of Sciences, 995 (2003), [7] Y. Dor, V. Djonov and E. Keshet, Making vascular networks in the adult: Branching morphogenesis without a roadmap, Trends in Cell Biology, 13 (2003), [8] (MR ) A. Friedman, A multiscale tumor model, Interfaces and Free Boundaries, 10 (2008),

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