A model of wound-healing angiogenesis in soft tissue

ELSEVIER

A Model of Wound-Healing Angiogenesis in Soft Tissue G. J. PETI'ET, H. M. BYRNE, 1 D. L. S. MCELWAIN, AND J. NORBURY 2

School of Mathematics, Queensland University of Technology, Brisbane QLD 4001, Australia Received 19 January 1995; revised 29 November 1995

ABSTRACT Angiogenesis, or blood vessel growth, is a critical step in the wound-healing process, involving the chemotactic response of blood vessel endothelial cells to macrophage-derived factors produced in the wound space. In this article, we formulate a system of partial differential equations that model the evolution of the capillary-tip endothelial cells, macrophage-derived chemoattractants, and the new blood vessels during the tissue repair process. Chemotaxis is incorporated as a dominant feature of the model, driving the wave-like ingrowth of the wound-healing unit. The resulting model admits traveling wave solutions that exhibit many of the features characteristic of wound healing in soft tissue. The steady propagation of the healing unit through the wound space, the development of a dense band of fine, tipped capillaries near the leading edge of the wound-healing unit (the brush-border effect), and an elevated vessel density associated with newly healed wounds, prior to vascular remodeling, are all discernible from numerical simulations of the full model. Numerical simulations mimic not only the normal progression of wound healing but also the potential for some wounds to fail to heal. Through the development and analysis of a simplified model, insight is gained into how the balance between chemotaxis, tip proliferation, and tip death affects the structure and speed of propagation of the healing unit. Further, expressions defining the healed vessel density and the wavespeed in terms of known parameters lead naturally to the identification of a maximum wavespeed for the wound-healing process and to bounds on the healed vessel density. The implications of these results for wound-healing management are also discussed.

1Current affiliation: Mathematics Department, UMIST, Manchester, M60 1QD, United Kingdom. 2Current affiliation: The Mathematical Institute, Oxford University, 24-29 St Giles, Oxford OX1 3LB, United Kingdom.

MATHEMATICAL BIOSCIENCES 136:35-63 (1996) © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010

0025-5564/96/$15.00 PII S0025-5564(96)00044-2

36

G.J. PETTET ET AL.

1. INTRODUCTION In recent years considerable interest in modeling the processes involved in wound healing has been shown. A number of models have been concerned with the healing of wounds in the mammalian epidermis, concentrating specifically upon the mechanisms involved in the regulation of epidermal cell mitosis [38-40] or upon the interaction of fibroblasts and the extracellular matrix in the epidermis [44, 45]. Such models may be termed avascular as they do not need to directly involve the role of blood vessels. In contrast to epidermal wound-healing models, we consider in this article a mathematical model of wound healing in the soft tissue of the dermis. Here the process of angiogenesis or new blood vessel growth is intimately involved, as is the new vasculature that acts as a source of vital nutrients and a sink of waste products for the developing granulation tissue. In addition to its importance in soft-tissue wound healing [2, 9, 10, 12, 18], angiogenesis also has an integral role in other growth and repair processes such as embryogenesis [1] and tumor growth [16, 17, 31, 35]. The large body of research that has been developed in the study of the mechanisms involved in angiogenesis is testimony to its importance in biological systems. Of particular interest in this article are observations made of the healing of small wounds in the soft tissue of rabbit ears, which have led to the identification of a well-defined sequence of events that characterize the ingrowth of new blood vessels [2, 9, 10]. A dominant feature of a healing wound is the healing unit or wound module [2], which is composed of a leading front of fibroblasts, closely followed by a band of fine, tipped capillaries and a trailing network of maturing vessels. The motion of this unit defines the progress of healing. Experimental results suggest that the growth of this structure may be controlled by a feedback mechanism involving tissue oxygen concentration and macrophage-derived angiogenesis factors [2]. At the low oxygen concentrations typical of an enclosed wound space, the macrophages, which are recruited to the wound site in an early response to wounding, secrete a number of chemicals. These macrophage-derived chemicals, together with a number of other chemical factors produced in the early inflammatory response to wounding, elicit various responses from the nearby tissue cells. Some act as mitogens triggering cells to multiply, others stimulate motility (chemokinesis), while others direct the motion of migrating cells (chemotaxis) [4]. In this article, for simplicity, we do not distinguish between these various role-specific chemical agents and so use the generic terms angiogenesis factors or chemoattractants to describe the macrophage-derived chemicals.

WOUND-HEALING ANGIOGENESIS

37

The ingrowth of new blood vessels, driven by the macrophage-derived angiogenic chemicals, begins with the development of precursors to new vessels, termed buds. The formation of buds on the intact vessels, near the wound margin, is induced by elevated levels of angiogenic factor. Each bud extends to form a short sprout whose tip moves in the direction of the wound center, that is, toward the source of the angiogenic stimulus. As the sprout, or capillary tip, migrates it leaves, in its wake, a new capillary that is contiguous with the parent vessel. Preceding the new vessel is a background population of tissue cells and fibroblasts, which together with the vessel cells, constitutes the living portion of the repair tissue. While the tipped vessels, or capillaries, are traveling toward the wound center there are many tip-to-tip and tip-tovessel contacts in which, for example, two tipped vessels join to form a single loop, or arcade, in a process termed anastomosis. The increase in the local capillary density causes the local oxygen concentration to rise. Macrophages at the wound site respond to changes in the local environment in the following way: As the oxygen concentration rises, the rate of production of the angiogenesis factor falls, eventually ceasing altogether. Thus, depletion of the angiogenic/chemotactic factor, by mechanisms such as natural decay and extraction by the vasculature, eventually dominates and prevents further capillary bud formation [8, 25, 26, 27, 34]. In this article no explicit mention is made of either the macrophage population or the oxygen tension. Instead we formulate the roles they play in the wound-healing process in terms of the local vessel density. In particular, we assume that low levels of vessel density indicate poorly oxygenated areas that are rich in active, attractant-producing macrophages, whereas higher vessel densities correspond to wel-oxygenated tissue with inactive macrophages. We note that vessel densities in the newly repaired tissue are typically much higher than those of normal or undamaged tissue. Once repair of a portion of the wound is complete, these high vessel densities are reduced to levels of normal tissue through a process of vessel remodeling [21], where morphological changes in the blood vessels, such as endothelial shortening and apoptosis or natural cell death, effect the gradual reduction of the vessel density [37, 46]. The remodeling causes a retraction of the vessel, rather than a fragmentation and degeneration. Thickening and shortening of the new capillary loops is also characteristic of the maturing vasculature. The remodeling phase typically lasts for several months, or even years, a time scale of much greater magnitude than that required to effect the closure of a wound (typically about 10-14 days) and, as such, is not considered explicitly in this article. The self-regulatory mechanism described above, involving oxygen as

38

G. J. PETTET ET AL.

a modifier of attractant production, is the main feature that distinguishes angiogenesis in wound healing from angiogenesis in pathologies such as tumor growth [16, 17, 31, 35]. A further distinguishing feature is the manner in which the vascular remodeling that occurs in the later stages of wound-healing angiogenesis restores the vessel density to a level near "normal" for the type of tissue involved [11]. In contrast, in tumor angiogenesis the "final" vascular density may be considerably higher than that of normal tissue. It is anticipated that, with minor modifications, the model of angiogenesis presented in this article might well be applicable to other angiogenic environments, such as tumor angiogenesis, diabetes, and arthritis. To summarize, in this article we focus on angiogenesis associated with soft-tissue wound healing. Through the development and analysis of a simple mathematical model, we aim to elucidate the relative importance of a number of the different processes known to operate within the angiogenic environment. The resulting model comprises a system of partial differential equations, which describe the evolution of the capillary-tip endothelial cells, the macrophage-derived chemoattractant, and the new blood vessels. To minimize the complexity of the model and to maintain our focus on the developing vascular front, no explicit mention is made of the macrophages or the oxygen tension for reasons outlined previously. The roles played by the fibroblasts and extracellular matrix are also not explicitly described in our model as they are not seen as being crucial to the process of angiogenesis. Rather, they may have a modulatory influence upon endothelial cell chemotaxis in the sense that the extracellular matrix provides sites of attachment for the migrating capillary-tip cells and the fibroblasts may have an endothelial cell proliferation inhibitory role as well as acting as a source of extracellular matrix [22, 33]. The roles of these variables in the healing process are formulated in terms of the capillary-tip endothelial cells, the macrophage-derived attractant, and the blood vessels. Chemotaxis drives the wave like ingrowth of the wound-healing unit and is incorporated as a dominant feature of the model that admits traveling wave solutions capable of exhibiting many of the characteristic features of wound healing in soft tissue. For example, numerical simulations of the full model exhibit the steady propagation of the healing unit through the wound space and the development of a dense band of fine, tipped capillaries near the leading edge of the wound-healing unit (the brush-border effect). Further simulations show that the full model can exhibit behaviors departing from that of normal healing. Given the appropriate choice of parameter values, failure of the wound to heal is also a possible steady state of our full model, thus mimicking pathological conditions such as ulcers. In addition, the elevated vessel density

WOUND-HEALING ANGIOGENESIS

39

associated with newly healed wounds, prior to vascular remodeling, is discernible. The self-regulatory nature of angiogenesis in wound healing is also observed and distinguishes our model from other mathematical descriptions of angiogenesis in, for example, tumors [3, 6, 7]. Through the development and analysis of a simplified model, where the simplification is determined by the relationships between the magnitudes of the parameters used in the numerical simulations, approximate forms for the wavefront are constructed and valuable insight is gained into how the balance between chemotaxis, tip proliferation, and tip death affects the structure and the speed of propagation of the healing unit. In particular, expressions defining the healed vessel density and the wavespeed in terms of known parameters lead to the identification of a maximum wavespeed for the wound-healing process and to bounds on the healed vessel density. The implications of these results for wound-healing management are also discussed. 2. THE MODEL In this article we focus on the experimental model system as used by Cliff [9, 10] and others [25-27]. The experimental configuration involves a thin disc-like wound chamber within the dermis of a rabbit's ear. The chamber is sealed on the upper surface by a transparent membrane, so that observation, by microscope, may be made of the progress of healing as demonstrated by the ingrowth of new tissue from the chamber margin toward the wound center. A good description of the rabbit ear wound chamber may be found in [25, 46]. We remark that this configuration corresponds to an approximately two-dimensional wound. Following Balding and McElwain, who modeled tumor-induced angiogenesis [3], the continuum model of soft-tissue wound healing we propose is based on the fungal growth model of Edelstein et al. [14, 15], the three processes sharing a number of common features, which include branching, anastomosis, and migration. However, as mentioned in the introduction, angiogenesis in wound healing is governed by a regulatory feedback loop that limits the vessel density in the newly healed wound. This feature distinguishes our model from other models of angiogenesis [3, 6, 7]. Angiogenesis is essentially a multidimensional process, with new tips sprouting in directions other than that of the advancing front. By assuming radial symmetry and by averaging the dependent variables perpendicular to the direction of motion of the healing unit, we approximate Cliff's two-dimensional wound by a one-dimensional model, the spatial variable connecting the wound center to the wound edge, and formulate our model in one-dimensional Cartesian coordinates (this

40

G.J. PETTET ET AL.

describes the healing of a long, thin, rectangular surface wound in two dimensions). We introduce independent variables t and x, where t denotes time and the x-axis lies parallel to the direction of tip growth, such that the wound center lies at x = L and the initial wound margin at x = 0 . A schematic diagram of the one-dimensional (spatial) system modeled in this article is given in Fig. 1. Three averaged dependent variables are used to describe the relevant physical processes: capillary-tip density n(x, t), which we interpret as the number of tips per unit cross-sectional area in the plane perpendicular to the direction of wound front propagation; the chemoattractant concentration a(x, t), which we interpret as the mean concentration of chemoattractant per unit cross-sectional area; and the blood vessel density b(x,t), which we interpret as the vessel area per unit cross-sectional area. We note that no distinction is made between primary and secondary vessels or between new and old vessels; that is, we assume that all vessels are of equal diameter and permeability. We propose the following system of partial differential equations to describe the evolution of the dependent variables n, a, and b: an

a

at aa at

ax (Jn)+ fn(n'a'b'nx'ax'bx)' a ax (Ja)+ f~(n'a'b'nx'ax'bx)'

ab at

a ax (Jb)+ fb(n'a'b'n~'a~'bx)'

where Jn, Ja, and Jb are the flux of tips, chemoattractant, and blood vessels respectively, while fn, fa, and fb are kinetic terms, specified below. Our formulation for the tip flux Jn has two components. The first diffusion-like component represents the contribution to J~ from the observed random motility of the tips. This effect enables ostensibly stationary endothelial cells to occasionally extend pseudopods and to reposition themselves among the extracellular matrix and neighboring cells. On the basis of experimental results [42, 43] the second tip flux component describes the active migration of the capillary-tip cells up gradients of the macrophage-derived chemoattractant. Specifically, we adopt the following form for the tip flux, J ~ = - t x ~ ( a n / a x ) + x n ( a a / a x ) , where /x~ and X are the assumed constant coefficients of random motility and chemotaxis, respectively. This formulation for a chemotactically induced component of flux has been used by many other authors [7, 19, 24, 29, 30, 43]. We anticipate that relative to

WOUND-HEALING ANGIOGENESIS

41

(/} 4~,

O)

o

2

iT_

=E

~.=

O

tel.o

O

~mO

O

=t

.o

Q,I U

eO

t,,.,

"0

'~ ~

0

m

~o~

0 it)

0

L.L

o=~

G. J. PETrET ET AL.

42

chemotaxis, the contribution to the tip flux from the random motility is negligible. Assuming that the flux of attractant is due to random motion, we specify Ja = - D ( c ~ a / O x ) , where D is assumed constant. Following our expression for Jn, a contribution due to random motility of the blood vessels is included in our expression for the flux of blood vessels Jb" To reflect the fact that the vessel motion being modeled here takes place as a result of tip motion "dragging" the vasculature along, we modify the usual Fickian diffusion to include a dependence upon tip density, giving Jb = --tzbn(Ob/Ox), and assume that the constant vessel random motility coefficient /zb is of magnitude similar to that of/zn. The kinetic term fn associated with the tips comprises a production and a loss term. Taking cognizance of the fact that tip proliferation arises from secondary branching, with tips emanating from the blood vessels and stimulated by the presence of the angiogenic factor, we postulate that tip production or budding is proportional to the product of the attractant concentration and the vessel density, with constant of proportionality A~. For simplicity, we assume that natural death and tip-to-tip anastomosis constitute the dominant contributions to tip loss, and we neglect other cell loss mechanisms such as tip-to-branch anastomosis. Natural death is modeled by a linear decay of tips, with rate A2, while tip-to-tip anastomosis is modeled using a term proportional to the square of the tip density, with rate )to. Combining the above, we have f,, = A~ab- A.zn- A0n2 SO that the tip density satisfies the following equation:

0n O----~ ~

/'~n

02n

O~x 2 -random motility

O ( On -a- ~)

X"~

chemotaxis

_{_

Alab budding

-

A2n decay

-

A0n2"

(l)

anastomosis

We now develop fa, the kinetic term associated with the chemoattractant. The attractant is produced by macrophages that are located within the wound space, the local oxygen concentration regulating the rate of attractant production. By assuming that the wound space contains an abundance of macrophages--in particular, we assume that the macrophages are distributed uniformly throughout the wound space - - a n d that the oxygen concentration is proportional to the local blood vessel density, it is possible to omit explicit mention of the macrophages and oxygen concentration in our model. Such an assumption is reasonable when the oxygen diffusion time scale is considerably smaller than the time scales associated with the other species involved. We assume that deep into the wound, where both the vessel density and the oxygen concentration are low, attractant production occurs at the maximum rate A3. Toward the wound margin both the vessel density and the

WOUND-HEALING ANGIOGENESIS

43

oxygen concentration increase and we assume that the rate of attractant production falls monotonically to zero. Denoting a typical vessel density at which attractant production is negligible by bchar, w e assume that the rate of attractant production is given by (A 3/2)(1 + tanh((bchar - b)/6)), where the constant 6 determines the rate at which attractant production falls with increasing vessel density. In addition, we assume that natural decay and removal via the local vascular network are the dominant mechanisms for attractant loss and that these processes occur at rates /~4 and h 5, respectively. Combining the above yields f, = ( A 3 / 2 ) ( 1 + tanh((bchar - b ) / 6 ) ) Aaa- As ab and the following governing equation for the attractant concentration:

c~a 0"--7 =

D 02a

A3 ( 1 +tanh(bch"~-- b ) )

0X2 + ~ diffusion

A4a_Asab. -- decay

(2)

...... I

production

Again following [3], when considering the kinetic term associated with the blood vessels, we assume that the dominant method of vessel production is by so-called "snail trails": As a particular tip migrates the capillary behind it elongates at a rate that maintains contiguity of the vessel [14]. Thus as the tip moves a trail of new vessels is produced behind it at the rate -tzn(On/Ox)+ xn(Oa/Ox). Branch loss may be incorporated to describe the remodeling process by which the elevated level of vasculature associated with newly repaired wounds is reduced to a level close to that of undamaged tissue: Mechanisms of morphological change such as endothelial shortening and apoptosis lead to a gradual reduction in the vessel density. Since this process takes place over a longer period (approximately months or years) then the migration of the healing unit or vascular front (approximately weeks) it is not considered here, although it is included in other work in progress. In summary, the equation describing the evolution of the vessel density we propose is given by

O [ Ob ~ [ On Oa 1 O__bb= p.b --~ ~n --j-~) -- I ~" -ff-x - x n -j-~ ] Ot random motility snail trail

(3)

Working on x E[0,2L], t~[0,~), we now impose boundary and initial conditions that close the system of equations defined above. Symmetry of the wound about its center (x = L) enables us to restrict attention to the "half-wound" 0 ~
On ( L , t ) = O = ~Oa ( L , t ) = --~-~ Ob (L, t). 0x

(4)

44

G. J. PETYET ET AL.

At the wound edge, or intact margin (x = 0), we assume that, in an early response to wounding, the tip density attains an elevated value, ~ say, which decays exponentially to zero at the rate a [6, 35, 43]. This mimics observations that suggest that during the early stages of angiogenesis the intact mature vasculature around the wound is the site of new sprout or tip formation. As the healing process continues, with the migration of tips producing newly formed capillaries, tip formation from the intact preexisting vasculature ceases and is replaced by tip formation on the newly formed vasculature [28]. We assume that the vessel density at the wound margin is equal to that of the healed state, b. Thus we prescribe

n(O,t) = he -~t

and

b(0, t) = b.

(5)

Blood-borne removal of attractant from the wound is modeled by imposing the flux condition

D-9-~(Oa "0,t) = h7ba(O,t),

(6)

for some assumed constant removal rate A7. Assuming further that initially the wound margin has penetrated a distance ~ into the wound so that the half-wound extends from x = $ to x = L and has width L - ~, we impose the initial data

n(x,O)=(:o(X )

for x ~ [ 0 , ~ ) for x E [ ~ , L ] ,

a(x,O) = 0 Vx ~ [ 0 , L ] , [bo(x) for x ~ [0,~) b(x,O) ( f l = b 0 ( £ ) for x ~ [ £ , L ] ,

(7) (8) (9)

where /3/> 0 is an assumed initial uniform vessel density inside the wound, and L - ~ is the initial half-width of the wound. The parameter ~7 can be interpreted as delineating the inflammation zone that develops as a margin around the wound edge in an initial response to wounding [8, 9]. In the inflammation zone there is considerable disruption to the vasculature and so it is expected that both the tip density and the vessel density fall monotonically from their values of ~ and b at x = 0 to zero and /3, respectively, as x increases from zero to £. Before continuing it is helpful to recast the model in dimensionless variables, mapping the half-wound space onto the unit interval and rescaling time with the corresponding attractant diffusion time scale L:/D. Defining independent variables (x*,t*)=(x/L, Dt/L:), the

WOUND-HEALING ANGIOGENESIS

45

dependent variables and key parameters are rescaled as follows:

n*= Ln

a* = h4a

bchar '

b*---b

A3 '

//~* = /Zn

X*

xA3

ho Lbchar

O

A]'=

2 AsLchar A~ -~

O

=

AI h 3 L 3

'

n* '

6, = '

hL

O

h 4L 2

'

A]=

D

'

a* = otZ2

A7Lbchar A'~ = - - - - - D ~ '

= bchar ,

t~ bchar'

A2 L 2

DA4 , A~=

b bchar

/.t,bbchar

=

X~=

/~, =

bchar '

D "

Dropping the asterisks, the system (1)-(9) transforms to give in dimensionless form

On 02n 3(Oa) 0--7 = lXn Ox'-'-f-'-- - X-d-~ n--~ Oa Ot Ob

+ Alab - h2n

OZa + l+tanh ----7 Ox T 0 [ Ob ~ On

Aon2 ,

(10)

- ( A 4 + Asb)a,

(11)

-

Oa

0"-'7 = t'~b " ~ [ n -~X ) -- ~ n -'-~ + x n "-~'-~,

(12)

subject to

n(O,t) = he -~t,

-Oa ~ ( O,t ) = A7a(O,t)b,

On ( 1 , t ) = 0 --- Oa ax

n(x,O)={;o(X) a(x,O) = 0

=

,gb ( 1 , t ) ,

for x ~ [ 0 , £ ) for x ~ [£, I], Vx ~ [0,1],

(bo(x) b(x,O)= ([B=bo(~) 3.

(l't)

b(O,t) = b, (13) (14) (15) (16)

for x ~ [0,.~) for x ~ [£,1].

(17)

N U M E R I C A L RESULTS

The system of parabolic partial differential equations (10)-(12), subject to the boundary and initial conditions (13)-(17), was solved using the method of lines and Gear's method, as implemented by the NAG Fortran Library routine DO3PGF. Figures 2, 3, and 4 show solution profiles of the tip density n, the attractant concentration a, and the vessel density b for three sets of

46

G . J . PETTET ET AL.

(a)

W o u n d Centre

Wound Margin 1

~4 o .4 0 . 8

-> t -> t=6

t=30

~0.6 .,4 ~A ~0.2 g~4 0 0

0.2

0.4

Distance (b)

0.6

from Wound Margin

0.8

1

(dimensionless) W o u n d Centre

wound Margin

1.5i

~l.2S 1>,0.751-

~0"21[ '

0

.

.

.

.

.

.

.

.

.

.

0.2

Distance

.

.

,

.

0.4

from W o u n d M a r g i n

.

.

0.6

.

.

,

i

0.8

I

,

,

,

i

1

(dimensionless)

FIG. 2. P r o f i l e s o f t h e c a p i l l a r y - t i p d e n s i t y n, m a c r o p h a g e - d e r i v e d a t t r a c t a n t c o n c e n t r a t i o n a, a n d b l o o d v e s s e l c o n c e n t r a t i o n b p r o p a g a t i n g f r o m t h e w o u n d m a r g i n at x = 0, t h r o u g h t h e w o u n d s p a c e , to t h e w o u n d c e n t e r at x = 1. P r o f i l e s a r e p l o t t e d at t i m e s t = 0, 6 , 1 2 . . . . . 60, w i t h t i m e i n c r e a s i n g f r o m left to right. P a r a m e t e r v a l u e s : /z, = 0.001, /x b = 0 . 0 0 1 , X = 0.1, b = 1.5, b~ = O, Ao = 100, A 1 = 100, A 2 = 10, A4 = 100, A5 = 10, t~ = 1, A 7 = 10, 3 = 0.01, a n d a = 2.5.

WOUND-HEALING ANGIOGENESIS (n)

47 Wound Centre

Wound Margin

-> t ->

A0.8 t=6

t=30

-H ~0.6

! ~0.4

"~0.2

i

0.2

0.4

0.6

0.8

1

Distance from Wound Margin (dimensionless)

FIG. 2.

(Continued)

parameter values. In each case the initial width of the inflammation zone is given by ~--0.05 and the initial tip and vessel densities are given by h ( x - £ ) ( 2 x 2 - ~ c - ~2) b(x,O)=(~-/3)

n(x,O) = 0 ,

(x-fc)(2x2-fac-£2)+b b(x,O)=/~

for x ~ [ 0 , £ ) , for x ~ [0,£), for x ~ [.~,1],

where /~ is the assumed initial uniform vessel density inside the wound space. Where possible our choice of parameter values is based on estimates derived from experimental data. However, the sparsity and inconsistency of certain parameter estimates meant that several parameter values were selected to produce solution profiles that are in good qualitative agreement with experimental observations. In the absence of better parameter estimates, the quantitative predictions of the model are therefore limited. Notwithstanding the fact that parameter values determined from in vitro systems are likely to be different from those measured in vivo, we take: /zn = 10 -9 cm2/s as a typical motility coefficient for the microvessel endothelial cells that constitute the capillary tip; X = 10 3 cm2/s for an attractant concentration of approxi-

48

G.J. PETTET ET AL.

mately 10-I°M; and D = 10 - 6 c m 2 / s for the attractant diffusion coefficient [42]. Referring to the rescaling described in Section 2, we deduce that, in dimensionless form, D = 1, /'/'n 10-3, and X = 10-1 By insisting that the dimensionless attractant concentration be O(1) and by assuming a typical attractant concentration atyp = 10-~°M, we deduce further from the rescaling presented in Section 2 that the rates of attractant production (A 3) and attractant decay (A 4) are related in the following way: A.3 / / ) t 4 ~ aty p ~ 10-10. In addition, by defining L ~ 2.5 mm as a characteristic length inside the wound chamber [23, 32, 47] and by considering the magnitude of the parameters used in our simulations it is possible to derive estimates for other system parameters such as A1 (the rate of tip proliferation for a given attractant concentration and vessel density), A: (the rate of capillary tip "death"), and /~4 (the attractant natural decay rate). Thus validation of our model formulation may be found by comparing the above predicted values with experimentally determined estimates. Figure 2 shows the profiles of the capillary tip density, the attractant concentration and the vessel density at regular time intervals when successful angiogenesis and subsequent wound healing occurs. In this simulation the (elevated) vessel density within the initial inflammation zone is fixed at b = 1.5. We observe that, initially, as a result of the buildup of attractant, tips at the wounded edge start to proliferate, generating a surge in capillary tips in a neighborhood of the wound margin (see Fig. 2a), which in turn causes a rapid rise in the vessel density there (see Fig. 2c). Decay of the attractant together with its removal by the vessels then produces a spatial gradient in the attractant field (see Fig. 2b), which facilitates chemotactic motion of the tips into the wound. As the tips move into the wound space new vessels are produced to maintain contiguity of the developing vasculature. For t > 6, the model exhibits the wave-like ingrowth of capillary tips and new blood vessels characteristic of angiogenesis in wound healing. The qualitative features of the brush-border effect are observed as a narrow leading margin of densely packed capillary tips [9, 10, 36] propagating through the wound space in a wave-like manner, followed by the new blood vessels. Behind the healing front blood-borne removal of attractant enables tip decay and anastomosis to restore the elevated tip density in the healed tissue to zero. We note that, as time increases, the solution tends to a "healed" steady state for which n ( x ) = 0 = a ( x ) for all x ~ [0,1]. The absence of vessel remodeling in Eq. (12) means that the surge in vessel density b > b = 1.5 is still apparent in the healed state. The regular spacing of =

WOUND-HEALING ANGIOGENESIS

49

the solution profiles as t increases suggests that the vascular front propagates with a constant wavespeed. From Fig. 2 we estimate that the wavespeed of the invading brush border is 2 . 6 x 1 0 -2 (dimensionless units). With D = 10 - 6 c m 2 / / s the attractant diffusion time scale for a wound with characteristic length L = 2.5X 10 -1 cm (the radius of a rabbit ear wound chamber) is T = L2/D = 7.25x 10 4 S ~ 0.8 d a y . This gives an approximate healing time of 30 days, which is in good agreement with experimentally observed healing times [9, 10, 47]. Numerical simulations were conducted varying those parameters that influence the development of the capillary tip. When the chemotactic coefficient X was reduced to X = 0.025 the profiles for n, a, and b remained qualitatively similar to those depicted in Fig. 2 for which X = 0.1. However, the wavespeed c fell from 2.6×10 -2 to 9 . 7 x 1 0 -3 dimensionless units. As a consequence of the fall in c, the density of new blood vessels in the healed wound, although larger than the "normal" value for undamaged tissue (b = 1), reduced from a value of 1.65 to 1.25 with X equal to 0.1 and X = 0.025, respectively. Further simulations were performed, assigning the values used in Fig. 2 to all parameters, except for )to and A1, which were reduced from 100 to 10. The resulting solution profiles were again qualitatively similar to those shown in Fig. 2. Decreasing )to and )t1 led to reductions in the wavespeed and the healed vessel density, which were similar to those achieved by reducing X: c fell from 2 . 6 x 10 -2 to 1.4x 10-2; the healed vessel density fell from 1.65 to 1.4. Figure 3 shows the solution profiles when the tip production parameter A1 is set to zero, the tip loss parameters )to and A2 are set at 10 -1, and all other parameter values are the same as those used to generate Fig. 2. This simulation may be considered an example of impaired healing, where setting A1 to zero assumes that tip production has been disabled, perhaps by chemically blocking the mitotic activity of the endothelial cells constituting the vessels. Another scenario that might be represented here is that of the impaired ability of the vasculature of some wounds to form new capillary buds as a consequence of their encasement in "fibrin cuffs" [22]. The figure shows the arresting of the healing process once the initial flush of tips have begun to migrate into the wound space. In the absence of tip production, any tips present are removed by death and tip-to-tip anastomosis. As a consequence of the limited invasion of capillary tips, the new vasculature is poorly developed, and the attractant rapidly attains a (spatially nonuniform) steady state. The profiles shown in Fig. 4 correspond to a situation in which the vasculature is only partially disrupted by an injury. This might be the

G. J. P E T F E T E T AL.

50

(a)

Wound Margin

Wound Centre

1

o

-~ o . 8 .,-t ~0.6 .,.t ~0.4 o ~0.2

0

°

.

.

.

.

,

0

.

.

.

.

°

0.2

.

.

.

.

0.4

°

.

.

.

.

0.6

,

.

.

.

.

0.8

,

1

Distance from Wound Margin (dimensionless)

(b)

Wound Margin

Wound Centre

1.4 ~1.2

o .,4

~o.8 "~ 0 . 6

~ 0.4 0.2 |

0

,

,

.

.

!

0.2

,

.

.

.

.

.

0.4

.

.

.

°

0.6

.

.

.

.

.

0.8

.

.

.

.

J

1

FIG. 3. Profiles of n, a, and b for a simulation of impaired healing where the propagation of a steady front of capillary tips delineating the wound-healing unit is hampered by the absence of new capillary sprout formation despite the presence of significant levels of chemoattractant. Profiles are plotted at times t = 0,3,6 ..... 30, with time increasing from left to right. All parameter values are the same as those in Fig. 2 with the exception of it 0 = 0.1, h I = 0, and h 2 = 0.1.

WOUND-HEALING (11)

ANGIOGENESIS

51

wound Margin

Wound Centre

~0.8

"d .o

.~ 0.6 "0 "~ 0.4,

BO.2

. . . . .

.

0.2

.

,

. . . .

0.4

, 0.6

. . . .

, 0.8

1

Distance from Wound Margin (dimensionless)

FIG. 3. (Continued) case after a deep graze to the skin. Once again we use the same parameter values as those used to generate Fig. 2 except that now we set b = 0.25 to reflect the partial disruption to the vasculature. The solution profiles for n, a, and b are qualitatively similar to those of Fig. 2. We note that, in comparison with the results shown in Fig. 2, with a nonzero vessel density in the wound space, the wavespeed of the healing front has increased to 3 . 2 x 1 0 -2 dimensionless units whereas the vessel density of the healed state has fallen to 1.36. Comparing Figs. 2, 3, and 4 we observe that once the attractant field has been established, the vessel and tip density profiles propagate through the wound space as traveling waves, with constant wavespeed. In addition, the vessel and tip densities are approximately constant both ahead of and behind the healing front. These observations are exploited in the next section where a simplified model for wound healing is developed. 4.

C H E M O T A C T I C B O U N D A R Y L A Y E R ANALYSIS

In this section we seek traveling wave solutions to the model for the asymptotic limit (/~0, /~1, /~4)~

~2 ('~O, ~1, ~4),

..~ : 8X,

(~.£n,~£b)~83(~)bn,~b), (18)

52

G . J . P E q T E T ET AL.

(a)

Wound Margin

Wound Centre

~o.6

i0.4 0.2

i

0

~

. . . .

0~

'

'

'o14

. . . .

0'.6 . . . . . .o.s. . .

Distance from Wound Margin (dimensionless)

FIG. 4. Profiles of n, a, and b showing the m a n n e r in which the presence of a small, nonzero vessel density in the wound space affects the healing process. Profiles are plotted at time t = 0 , 3 , 6 ..... 30, with time increasing from left to right. All parameter values are the same as those in Fig. 2 with the exception of b® = 0.25.

WOUND-HEALING ANGIOGENESIS (n)

53

Wound Centre

Wound Margin

~0.8

"i0.6 .~0.4 "~0.2

.

.

.

.

.

.

0.2

.

.

.

.

.

.

.

.

.

0.4

.

0.6

.

.

.

,

.

.

0.8

.

.

.

1

Distance fromWoundMargin (dimensionless) FIG. 4.

(Continued)

where 0 < e << 1 is a small parameter. With e ~ 0.1 this asymptotic limit approximates the parameter values employed in Figs. 2, 3, and 4. Referring to Fig. 2, we remark that, for the asymptotic limit (18), the establishment of an initial attractant field is rapid and that once this field is established the healing unit propagates as a traveling wave across the wound space, connecting the healed and wounded configurations. Further, this propagating wavefront persists until the healing unit reaches x = 1. From Fig. 2 we note that while the healing unit travels from the wound edge to the wound center the dynamic variation of the solution profiles is restricted to a neighborhood of the traveling front of the healing unit whose width is ~ 0 ( 6 ) . When transforming the governing equations to traveling wave coordinates we effectively pose our model on an infinite domain, an approximation that limits the type of comparisons that can be made between the numerical simulations presented in Section 3 and the asymptotic solutions derived below. However, the qualitative nature of the solution profiles are retained. We now transform the model from the original independent variables to the traveling wave coordinate z = x - e~t, where c = e? is the assumed constant wavespeed, to be determined as part of the solution. Introducing dependent variables defined by

N(z) =n(x,t),

A(z) =a(x,t),

B(z) = b ( x , t ) ,

54

G.J. PETI'ET ET AL.

using primes to denote differentiation with respect to z = x - eE't, and dropping the overbars, equations (10)-(12) transform to give - e 3 c N ' = esjz,,N " -

e 3 x ( N A ' ) ' + A1AB - 22A2N -

- , ~ 3 c A ' = e2A"+ A4H(1 - B) - A4A - e2AsAB,

- c B ' = e2tZb( NB') ' - 82tznN' +

xNA',

A0N2,

(19) (20) (21)

subject to the conditions U ' ( _+oo) = A'( _+~) = B'( _+oo) = 0.

(22)

We have used the scaling defined by (18) and have introduced b +~ to be the vessel density of the wounded and healed states, respectively, assuming that 0 ~
0 = AI.~¢._~- ho.///2, o = a,,,,, + , h H ( 1

- c ,.~n = X./Fagn .

-

-

(24)

(25)

WOUND-HEALING ANGIOGENESIS

55

From Eq. (23) we remark that, within the boundary layer the dynamics of the tip density are dominated by tip production and anastomosis. Equations (24) and (25) imply that the attractant distribution is determined by a balance between diffusion, production, and natural decay, while chemotactic motion of the tips determines the rate of production of the vessels. Introducing the parameter A

X //~'1

A=-~VTo

,

(26)

and substituting with X from (23) in (25), we obtain the third-order system of ordinary differential equations 0 = ~¢'n, + A a H ( 1 - ~ ) -/~4.j~¢'

(27)

0 = '-~n + 3A(5¢"~) 1/2"~¢~'

(28)

subject to ( s ¢ , ~ ' ) ~ (0, b 2)

as rt --* - ~ ,

(29)

(~,~)

as ~ --*~,

(30)

~ (1,b=)

these being the boundary conditions that ensure that the inner solutions match the outer solutions. We now deduce, by contradiction, that b_~ > 1. If b_~ ~<1, then, since b= < 1, by continuity in the wounded state, ~q~~<1 Vrl, giving H ( 1 - ~q~)- 1 in (27). Thus, no bounded solution connecting the outer solutions at 7/= + oo exists. With b_~ > 1 and fixing ~ ( ~ = 0 ) = 1 by a suitable choice of the traveling wavefront, the system partitions into the two regions, ~7~ (-0%0] and ~ [ 0 , ~ ) , with continuity of ( ~ ' , ~ ' , , ~ , ~ ¢ n ) imposed at 77=0. Solving (27), subject to the matching conditions ~¢( - o~) = 0, ~ ( ~ ) = 1 and the continuity condition for ~¢ at ~7= 0 but relaxing, as a consequence of the attractant production function H(1 - ~'), the smoothness condition for ~¢, at 7/= 0, we obtain the following expression for the attractant concentration inside the healing front: ,;go e o "

e~¢(') = [ 1 + ( ~ 0 - 1) e - ° '

for ~ E ( - 0%O] for ~ e [0, o~),

(31)

56

G.J. PETTET ET AL.

where O = ~ 4 and ~¢0 is to be determined. Equation (28) yields the following expression for the vessel density in terms of the attractant: 0 = d ~ {,.,~1/2 q._Aj~¢3/2},

(32)

=~ ,.~ = (A(1 -,3~¢3/2 ) -q- bl/2) 2,

(33)

having applied the matching condition as r / ~ oo and where the continuity of ~' across r/= 0 follows automatically from the continuity of d. Applying the condition ~q~(0)= 1 yields

=

Ix 1 j.)2J. X

"

Using (23), the tip density inside the boundary layer may be determined, J=

s¢~',

(34)

and, consequently, the tip density automatically satisfies the outer boundary conditions. In Fig. 5 the traveling wave profiles for X, ~¢, and ~' are plotted for typical values of the system parameters, such as those used in the simulation depicted in Fig. 2. Other traveling wave profiles for ~ , ~¢, Travelling Wave Profiles

1.5

N

tips

A

attractant

B

sprouts

~ 1.25 o - - -

o~

l

I/_2__

0.75

~o 0.5

0.25

e~

0

|

-1.5

,

-i

-0.5

0

0.5

1

1.5

Dimensionless Travelling Wave Co-ordinat.e

FIG. 5. Traveling wave profiles, traveling from left to right, for A r (--), .a¢ (---), and oq~ (---) obtained using the ehemotactic boundary layer model. Parameter values as per Fig. 2 using the wavespeed calculated from Fig. 2.

WOUND-HEALING ANGIOGENESIS

57

and ~ ' may be generated using nonzero values for b~ giving solutions in good agreement with those generated numerically as in Fig. 4. The solutions (./r,s¢, ~') obtained above are the leading-order terms of the asymptotic expansions inside the boundary layer, valid to 0(8). By considering higher-order terms for Eqs. (23)-(25), it would be possible to increase the accuracy of the above expansion. Having constructed approximate solution profiles, the boundary conditions ~,(-oo) = b_~ and ~ ¢ ( - ~ ) = 0 supply an expression that defines the vessel density of the healed state b_.,. in terms of the vessel density of the wounded state b.,. together with the parameters X, A1, )to and the wavespeed c. In particular, we find

b

=(A +b J/2) 2,

(35)

which together with Eq. (26) gives

1 c =

bV~

-

) b~/2

"

(36)

From the above results we might deduce that the rate of healing can be manipulated, not without, however, influencing the density of the healed state. Specifically, the wound-healing time can be reduced by increasing either X or A1 and by decreasing A0. As is not unreasonable to expect, there is some discrepancy between the numerical solutions of the previous section that apply to the entire domain of the wound space and the solutions we have derived here for within the chemotactic boundary layer. In particular the numerical solutions suggest that for the healed vessel density b_~ and the wavespeed c the dependence upon X, A1, and A0 implied by the numerical solutions of the full system is not the same as the dependence upon the ratio A~/)t o as shown above. Notwithstanding these discrepancies, the solutions within the chemotactic boundary layer do allow us to draw some conclusions about the behavior of the full system. For valid solutions within the chemotactic boundary layer we require 0 ~
(37)

and C < Cma x =

~

.

(38)

58

G.J. PEqTET ET AL.

For reasons of continuity, the upper bound for c developed from the chemotactic boundary layer analysis may also be used to bound the wavespeed for solutions of the full system. By measuring the wavespeed and the system parameters under different experimental conditions it should be possible to validate our model. Such experiments would have to satisfy the assumptions inherent in our model; for example, the timeframe to which the model is applied must be such that the neglect of vascular remodeling is valid. 5.

DISCUSSION

In this article we have developed a continuum model that quantitatively describes the sequence of events associated with angiogenesis in soft-tissue wound healing. It is significantly different from: (i) models describing epidermal wound healing [38-41] in which angiogenesis is not inw~lved; (ii) models describing tumor angiogenesis [6, 7]; and (iii) stochastic models describing angiogenesis [5, 13]. Our model focuses on the evolution of the capillary-tip endothelial cells, as they migrate in the direction of increasing macrophage-derived chemoattractant. Rapid production of new capillary endothelial cells ensures that the capillary tips retain contiguity with the parent vasculature: The new cells constitute the new vessels and fill the space that would otherwise be created as the tips move toward the wound center. This vessel-extension mechanism gives the impression of a leading, nonproliferating tip traveling into the wound space leaving behind a wake or snail trail of new vessels. The model we have proposed to describe this phenomenon includes a self-regulatory mechanism that controls the rate at which new cells migrate into the wound space and relates the production of chemoattractant to the density of the new vessel architecture, and so by implication to the local oxygen concentration. If the local vessel density increases beyond a critical level, the macrophages cease production of chemoattractant because of the associated rise in the local oxygen concentration [25-27]. When attractant production ceases the chemoattractant gradient falls and consequently the rate of capillary-tip invasion and vessel production are also diminished. This enables the oxygen level to fall sufficiently to restimulate attractant production, and ultimately to enhance capillary tip invasion. In proposing that this control mechanism dominates the rate of wound healing, our model enables us not only to suggest which physical mechanisms determine the characteristic features of the wound-healing unit, but also to indicate experiments that could be performed to validate our model. Numerical simulations show that our model is able to reproduce many of the qualitative features of wound healing in soft tissue. For

WOUND-HEALING ANGIOGENESIS

59

example, the steady propagation of the healing unit through the wound space and the evolution of a developed vascular network at the leading edge of the healing unit (the brush-border effect) are both discernible from the numerical simulations (see Figs. 2-4). Moreover, realistic wavespeeds and healing times can be estimated from the numerical simulations. The elevated vessel density associated with newly healed wounds, prior to vascular remodeling, is also observed. Furthermore, making appropriate choices for the parameters associated with tip production and tip loss we are able to produce numerical solutions that mimic the establishment of a wound that fails to heal where the failure is attributable to a restriction of the process of new capillary bud formation. Through the development and analysis of a simplified model, insight is gained into the manner in which the balance between chemotaxis, tip proliferation, and tip death affects the structure and the speed of propagation of the healing unit. The simplified model is derived by focusing on the mechanisms that operate within the healing unit, the narrow band occupied by the wavefronts of the numerically determined traveling wave profiles. By viewing the leading edge of the wavefronts as a chemotactic boundary layer, our analysis yields asymptotic solutions for capillary-tip density, chemoattractant concentration, and sprout or vessel density. Expressions defining the healed vessel density b_~ and the wavespeed c, in terms of the wounded vessel density b~ and other system parameters are derived and shown to be in good agreement with the numerical results. These relationships lead naturally to bounds on the healed vessel density and to the identification of a maximum wavespeed for the wound-healing process. In particular we deduce that C ~< Cma x : ( X / 3 ) V / ~ I / , ~

o.

One of the weaknesses of the continuum model presented here is its inability to describe how changes in the microstructure of the vasculature, such as thickening of the branches with time, effect the delivery of oxygen to the wound and surrounding tissue. In our model we have made no explicit mention of the oxygen concentration and have instead assumed that it is proportional to the vessel density. It is likely that the rate of oxygen delivery from the younger more permeable vessels near the wound edge is higher than that from the more mature blood vessels in normal tissue. Additionally our model assumes that a reduction in tip density may be ascribed to either anastomosis or tip death, so that any improvement in the vascular network arising from anastomosis is neglected. Some of these problems are due to the fact that we have modeled angiogenesis as a one-dimensional process whereas it is at least a two-dimensional process, with new tips and vessels forming in directions other than that of the propagating front. Improvements to

60

G. J. PEqTET ET AL.

the model that we are presently making include an extension to higher space dimensions together with a reformulation of the model to allow for the distinction between cell death and anastomosis. Furthermore, thickening of the branches and other changes to the vascular network that are direct consequences of anastomosis will be included, together with the longer time scale vascular remodeling. To elucidate the role played by the vasculature on the oxygen levels in the wound and, hence, upon the expression of angiogenesis factor by the wound macrophages we are also extending the model to include oxygen as a fourth species. Other extensions currently under investigation are the inclusion of fibroblasts and the extracellular matrix, which is generated during the wound-healing process [22, 20]. The outcome of our chemotactic boundary layer analysis has some implications for the management of soft-tissue wounds. The relationship between c and b_~ described in Eq. (36) suggests that it should be possible to manipulate the rate of healing by increasing either X or A1, that is, by enhancing the chemotactic response or by stimulating the vessel endothelial cells to increase the rate of capillary sprout budding or by decreasing )to, thus reducing the rate of tip loss by anastomosis. It is unlikely, as the numerical simulations suggest, that an increase in wound-healing speed could be achieved without affecting the vessel density of the healed state, b_~. The numerical solutions suggest that an increase in wavespeed (or reduction in healing time) is associated with an increase in the vessel density of the healed wound. It may be possible to verify this prediction by conducting experiments in which the rate of capillary-tip production for a given vessel density and attractant concentration is manipulated by enhancing or retarding capillary endothelial cell reproduction rates. Further experimental validation of our model of wound-healing angiogenesis may be provided by measuring the parameters A1 (the rate of tip proliferation for a given attractant concentration and vessel density), A2 (the rate of capillary-tip death), and A4 (the attractant natural decay rate), so that they may be compared with those predicted by our model. REFERENCES 1 B. Alberts, D. Bray, J. Lewis, M. Raft, K. Roberts, and J. Watson, The Molecular Biology of the Cell, Garland, New York, 1989, 3rd ed. 2 F. Arnold and D. C. West, Angiogenesis in wound healing, PharmacoL Ther. 52:407-422 (1991). 3 D. Balding and D. L. S. McElwain, A mathematical model of tumour-induced capillary growth, J. Theor. Biol. 114:53-73 (1985). 4 N.T. Bennet and G. S. Schultz, Growth factors and wound healing. II. Role in normal and chronic wound healing, Am. J. Surgery 166:74-81 (1993).

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61

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63

45 R.T. Tranquillo and J. D. Murray, Mechanistic model of wound contraction, J. Surg. Res. 55:233-247 (1993). 46 H.A.S. Van Den Brenk, Studies in restorative growth processes in mammalian wound healing, Br. J. Surgery 43:525-550 (1955). 47 D.F. Zawicki, R. K. Jain, G. W. Schmid-Schoenbein, and S. Chien, Dynamics of neovascularization in normal tissue, Microvasc. Res. 21:27-47 (1981).