Mathematical and Computer Sciences - Heriot-Watt Universityjas/paperpdfs/dallon3.pdf · 1999. 9. 17. · Mathematical and Computer Sciences - Heriot-Watt University

J. theor. Biol. (1999) 199, 449}471Article No. jtbi.1999.0971, available online at http://www.idealibrary.com on

Mathematical Modelling of Extracellular Matrix Dynamicsusing Discrete Cells: Fiber Orientation and Tissue Regeneration

JOHN C. DALLON*?, JONATHAN A. SHERRATT* AND PHILIP K. MAINI-

*Department of Mathematics, Heriot-=att ;niversity, Edinburgh, EH14 4AS, ;.K. and-Centre for Mathematical Biology, Mathematical Institute, ;niversity of Oxford, 24}29 St Giles1,

Oxford OX1 3¸B, ;.K.

(Received on 13 November 1998, Accepted in revised form on 18 May 1999)

Matrix orientation plays a crucial role in determining the severity of scar tissue after dermalwounding. We present a model framework which allows us to examine the interaction of manyof the factors involved in orientation and alignment. Within this framework, cells are con-sidered as discrete objects, while the matrix is modelled as a continuum. Using numericalsimulations, we investigate the e!ect on alignment of changing cell properties and of varyingcell interactions with collagen and "brin.

( 1999 Academic Press

1. Introduction

Alignment phenomena are found throughoutour environment, in contexts as varied as liquidcrystals (Priestly et al., 1975), intracellular actinnetworks (Small et al., 1995), "sh and insectswarming (Okubo, 1986), cellular biology (NuK -bler-Jung, 1987) and wound healing (Ehrlich& Krummel, 1996). In this paper, we focus onwound healing, where a higher degree of align-ment is a key characteristic of scar tissue. Anunderstanding of how processes interact to pro-duce alignment is a crucial step in the develop-ment of anti-scarring therapies.

Dermal skin tissue is composed of a relativelysparse cell population surrounded by a "brousnetwork of proteins called the extracellularmatrix. This matrix is primarily composed ofcollagen that interacts with cells in a dynamic

?Author to whom correspondence should be addressed.E-mail: [email protected]

0022}5193/99/016449#23 $30.00/0

way which is not fully understood (Hay, 1991). Itis clear that the extracellular matrix acts as a scaf-folding, providing directional cues via a phenom-enon known as &&contact guidance'' (Guido& Tranquillo, 1993; Clark et al., 1990; Hsieh& Chen, 1983). Fibroblasts, the cells which createand maintain the extracellular matrix, help toorient and give structure to the "brous network(Clark, 1996). This complex "broblast}extra-cellular matrix interaction, known as dynamicreciprocity, is the focus of this paper. We arespeci"cally concerned with the orientation of thecollagen matrix and learning what factors areimportant in alignment. The "broblasts in#uencethe collagen matrix both biochemically andmechanically. Evidence suggests that as the cellsmove and produce collagen the "brils areoriented in the direction of motion (Markwaldet al., 1979; Birk & Trelstad, 1986; Trelstad& Hayashi, 1979). Whether or not this is due totractional forces is unclear, although as "brob-lasts move they do exert tractional forces which

( 1999 Academic Press

450 J. C. DALLON E¹ A¸.

can alter the structure of the collagen matrix(Stopak & Harris, 1982; Harris et al., 1981). Inthis paper, for simplicity, we only consider align-ment processes which are associated with movingcells and e!ect the matrix near the cell, leaving aninvestigation of more long-range alignment andcontraction for future work.

Mathematical modelling of alignment has beenprevalent in the past few years, focusing on a widevariety of applications. Of particular relevance tothis study is the work on "broblast orientation(Edelstein-Keshet & Ermentrout, 1990; Mogilner& Edelstein-Keshet, 1995; Mogilner et al., 1996;Mogilner & Edelstein-Keshet, 1996); this focuseson orientation due to cell-to-cell contact inhibi-tion rather than directional cues taken from theirsubstrate, on which we concentrate. In contrastto this previous work, we are not primarily con-cerned with "broblast orientation as such, butrather on how this a!ects the orientation of the"brous network. The work which most closelyrelates to the modelling in this paper is thatwhich examines alignment of the extracellularmatrix (Olsen et al., 1998, 1999; Dallon & Sher-ratt, 1998; Barocas & Tranquillo, 1997). Each ofthese represents very di!erent approaches tomodelling similar systems. Olsen et al. (1998)model the system with reaction di!usion equa-tions where the species can take one of two ortho-gonal directions, while Olsen et al. (1999) usea related model including the long-range mech-anical forces within the "brous network. Dallon& Sherratt (1998) formulate the long-range angu-lar interactions as integro-di!erential equations,ignoring spatial variation, and in Barocas& Tranquillo (1997) the extracellular matrix istreated as a biphasic medium with viscoelasticproperties which both orients, and is oriented, bythe cells. All these models use continuum descrip-tions for variables, whereas the models describedin this paper represent the cells as discrete entitiesand the extracellular matrix as a continuumvector "eld.

In the next section we start our study bydescribing a simple model involving onlyone "brous matrix component (collagen). InSection 3, results from numerical simulations ofthe model are presented. Section 4 introducesa more complicated model which represents theextracellular matrix as two "brous networks, one

of collagen and one of "brin. By examininga simple model initially, and then a more com-plex extension, more is learned about the under-lying mechanisms for orientation. The results ofthe more complicated model are explained inSection 5 and we conclude with a discussion ofapplications to wound healing.

2. The Matrix Orientation Model

Initially, we only model simple interactionsbetween "brous collagen matrix and the "bro-blasts. Although we have a collagen network inmind, the model could be applied to any cellularreorganization of a "brous matrix. The cells aretreated as discrete objects whose paths are givenby f i (t)"( f i

1, f i

2), where the superscript denotes

cell i. The collagen is represented as a continuousvector "eld denoted by c (x, t) where x representsthe cartesian coordinates of a point in the plane.The vectors c are in R2, have unit length andtheir direction represents the predominant direc-tion of the collagen "bres at that point in spaceand time.

As mentioned previously, the cells receive di-rectional cues from the extracellular matrix.These cues are modelled by

df i(t)

dt,f0 i(t)"s

v (t)

DDv (t)DD(1)

with

v (t)"(1!o)c (f i (t), t)#of0 i (t!q)

DDf0 i (t!q)DD, (2)

where o and s are positive constants with s repre-sentating the cell speed and q a time lag. The "rstterm in eqn (2) represents the in#uence of thecollagen matrix on the direction of the cell (&&con-tact guidance'') and if o"0, the cell moves exact-ly in the direction of the collagen. Fibroblasts on"brous gels become very elongated and maintaina leading edge (Friedl et al., 1998) which wemodel by the second term of eqn (2) which givesthe cell a bias in the direction it is moving. Theparameter o determines the strength of this bias,so if o"1, the cell does not change direction andmoves in a straight line determined by the initial

FIG. 1. A graph of the weight function wi. The cell loca-

tion f i(t)"(0, 0).

c (x, t) evolves according to

dh (x, t)

dt"iDD f DD sin (a!h). (5)

MODELS OF EXTRACELLULAR MATRIX DYNAMICS 451

conditions, independent of its environment. Ineqn (1) the linear combination of directions isnormalized and scaled to give the appropriatespeed.

The "broblasts alter the collagen matrix bychanging its orientation. As mentioned pre-viously we model only the local #ux-inducedalignment. Since each "broblast can contributeto the reordering of the collagen, the overall e!ectof the cell population on the matrix is representedby the vector f

f (x, t)"N+

i/0

wi(x, t)

f0 i (t!q)

DDf0 i (t!q)DD, (3)

where q is again a time lag and N is the totalnumber of cells. This gives the cumulative e!ectof all the "broblasts on the collagen direction. Weassume that the e!ect of the cells acts in thisadditive fashion; this simplifying assumption isreasonable in view of the low cell density in oursimulations. The weight function, w

i, interpolates

the in#uence of the discrete cells to the con-tinuum collagen variable and maintains the in#u-ence of the "broblasts local. The issue of howdiscrete variables interact with continuum vari-ables is discussed in Dallon (1999) which dealswith numerical issues arising from discrete andcontinuum hybrid model formulations. Theweight function used here is graphed in Fig. 1 andis de"ned by

wi(x, t)"a

1a2

with

aj"maxG1!

D f ij(t)!x

jD

¸

, 0H, (4)

where x"(x1, x

2) and ¸"10 lm. Thus the sup-

port of the function is a 20]20 lm2.Ideally, the support of the weight function

should correspond to the shape of the cell, butsince it is constantly changing and unknown weuse the above simpli"cation. The time lag q rep-resents the time taken for the cell to changedirection after obtaining the directional cuesfrom its environment. For example, the head ofan elongated "broblast may be travelling in a dif-ferent direction than its tail. The orientation of

Here h is the angle of c, the vector representingthe collagen direction and a (x, t) is the angle off (x, t). Thus when the di!erence between theangles is small the derivative is small and whenthe directions are orthogonal the rate of changereaches a maximum. In addition, it is periodic sothat when a "bre and cell are oriented in eitherthe same direction or 1803 apart the "bre doesnot change directions.

A vector "eld representation of the "bres is notunique because the "bres are "laments whichextend in both the positive and negative directionof its associated vector. The representation of the"bres which we use is cell dependent and satis"es

Sc, f0 i(t)T!d'0. (6)

Here S), )T denotes the standard Euclidean innerproduct in the plane and d some small positiveconstant. We choose this representation to en-sure that a cell moves along the "bre in themanner which would require the cell to make thesmallest change in its direction. In other words, ifthe vector in the direction of the cell and thevector representing the "bre are placed head totail, the inner angle, /, is obtuse (see Fig. 2). If thecell direction and the "bre direction are at ap-proximately right angles (approximately de"nedby the choice of d), then the "bre orientation israndomly chosen to be the positive or negativedirection.

FIG. 2. A schematic depicting a cell direction and twopossible "bre orientations for the same "bre. Of the twopossible representations for the "bre orientation (a) and (b),only (b) satis"es eqn (6) for the shown cell direction. For thisrepresentation the inner angle / is greater than 903 whichmeans that when following the "bre, the cell alters its coursethe least rather than turning back on itself.

FIG. 3. The collagen orientations and cell positions fora typical simulation. In (a) the initial random collagen ori-entation is shown and in (b) the collagen orientation isshown after 100 hr of remodelling by the "broblasts ona domain of 0.5 mm]1.0 mm. The cells have a speed of15 kmhr~1, i"5, o"0, k"0.15 hr and the numerical gridfor the vector "eld is 51]101.


The numerical algorithm consists of the fol-lowing steps:

1. Interpolate the cell's direction to the grid.2. Orient the "bres with respect to the cell's

direction.3. Interpolate "bres to the cell's location.4. Normalize the interpolated direction.5. Change the "bres' direction and normalize.6. Move the cell in the direction found in

step 3.

Further details of the numerical scheme are givenin the appendix.

3. Results of Realignment

In this section, we describe results from themodel of collagen realignment by considering thee!ect of important parameters, the initial "breorientation, cell density, how cells enter thedomain and "nally di!erent boundary condi-tions. A typical simulation result is shown inFig. 3. There are 600 cells which start uni-formly distributed throughout the domain of0.5 mm]1 mm. The boundary conditions are re-#ective with the component of motion of a cell inthe direction perpendicular to the boundary en-countered being re#ected. This boundary condi-tion was chosen because it is the most realistic forthe epidermal boundary in the wound, can beinterpreted as a no-#ux boundary for the otherwound edges with an equal number of cells enter-ing and leaving the domain (for further details seethe appendix). The collagen orientation is repre-sented in the "gures by drawing curves whose

tangent corresponds to the direction of the vector"eld at the point, i.e. streamlines.

3.1. RATE AT WHICH THE FIBROBLASTS

ALTER THE COLLAGEN: i

The parameter i represents a cell's ability toalter the direction of the collagen "bres and isdetermined by properties of both the "broblastsand the collagen matrix. For instance, the sti!-ness of the collagen matrix, the local tractionalforces the cells create, the manner in which thecells produce collagen "bres and the degree thecells chemically alter the matrix will all in#uencehow easily the cells reorient the collagen. Of thesefour properties mentioned, only the tractionalforces of "broblasts have been measured, givinginsu$cient data for determining i. These forceshave been measured by placing cells on #oatingsilicon rubber "lms and measuring the distort-ment (Harris et al., 1980) and by measuring theoverall contractile forces on collagen gels withmoving "broblasts (Eastwood et al., 1996).

FIG. 4. The e!ect of altering the rate at which the "broblasts change the "bre direction. It is seen that as the in#uence of thecells on the collagen orientation increases the pattern has more structure in (a) where i"2.5, the most structure in (b) wherei"5 [same as Fig. 3(b)] and then less structure in (c) where i"20. The simulations shown here have the same parametersand set-up as that shown in Fig. 3.


Although this information is important, it is notsu$cient to determine i. Even though quantitat-ive values for i are not currently possible, we cancheck the model's sensitivity to i and later givesuggestions of how to roughly determine i ex-perimentally.

An important result of our numerical simula-tions is that the maximal smoothness of the ex-tracellular matrix occurs at a "nite value ofi (Fig. 4). That is, the degree by which a random-ly oriented vector "eld is smoothed, due to thecells, increases with i to a critical level and thendecreases. Large i means that the cells quicklyreorganize the collagen in the direction they aretravelling. The reduced alignment at large i oc-curs because the cells re-randomize the "eld anddo not create as much alignment.

The orientation of the "brous gel generated byour model can, when started with an initiallyrandom con"guration, remain random at aglobal level, yet have a lot of structure (seeFigs 3 and 4). Although globally there is nopredominant orientation, on a "ner scale di!er-

ent orientations will dominate resulting in localstructure and pattern. In order to compare di!er-ent simulations we want to quantify the amountof structure or alignment, but global averages areof little use. They cannot even distinguish be-tween the initial and "nal con"gurations. Thus,we use the following simplistic method tomeasure the structure which was inspired by thetechnique described in Rand & Wilson (1995).We subdivide the domain into uniform,non-intersecting regions. In each region a "bredirection is chosen and the absolute value of thedi!erences with all the other "bre directions with-in the region is averaged. This gives somemeasure for the alignment of the "bres within thesubregion. The values are then averaged for allthe subregions giving one value for the overalldomain. A priori, we do not know the scale of thestructure, so this procedure is carried out severaltimes with di!erent-sized subregions. We con-sider 2, 4, 8, 16, 32 and "nally 64 subregions. Theresults comparing the simulations with di!erentvalues of i are shown in Fig. 5. Of course, as the

FIG. 5. The measure of alignment applied to the results for simulations with di!erent values of i. The alignment is greatestfor i"5 and the corresponding alignment patterns are shown in Figs 3 and 4. The x-axis denotes the number of grid points inthe square subregions used. The symbols represent the values for i in the following manner: the random initial conditions (#),i"2.5 (} }]} }), i"5 (- -]- -) and i"20 (h).


regions become too small the measure becomesmeaningless since for smaller regions there arefewer points to average and in the extreme case ofa region with one grid point, the "bre is trulyaligned with itself.

One can see by the results that this crudemeasure con"rms what is visually obvious.Throughout the rest of the paper this measure ofalignment is used to con"rm comparisons whichare visually apparent and determine which simu-lations have more structure when a visual inspec-tion is not su$cient.

3.2. CELL SPEED: s

Increasing the speed of the cell has an obviouse!ect on the cell paths: they move farther and inlonger straight-line runs. Because the time step inthe simulations is kept constant, the cell samplesthe environment with the same frequency to de-termine its direction at higher speeds, but movesmore quickly between samplings. Similar resultscan be obtained by "xing the speed and alteringthe time step. A more subtle e!ect of variations incell speed is on the collagen alignment, and thisdepends on the support of the weight functionswi. If a cell moves su$ciently fast that the dis-

tance travelled in one time step is greater than thesupport of w

i, then at its new location the cell

does not continue to alter the "bres at its pre-vious location; we denote this critical value of thespeed above by s

w. Thus increases in speed do not

alter the extent of "bre alignment but simplyalign the "bres in a di!erent manner. At very lowspeeds the cell will alter its course many times ina small area causing little overall alignment in the"brous gel. As the speed increases the cell willalter its course fewer times in a given area butmodify the region within the support of w

icaus-

ing more alignment. Increasing the speed furtherwill decrease the e!ect since the cell modi"es the"bre at its previous positions to a lesser extent.When it moves faster than s

wit no longer

modi"es the previous position. Thus, we expectthere to be an optimal cell speed that aligns thegel to the greatest extent. This behaviour at lowspeeds is easily seen in Fig. 6 where an increasefrom 5 to 15 kmhr~1 clearly increases the align-ment of the gel. Yet as the speed increases theboundary conditions begin to signi"cantly a!ectthe patterns making it di$cult to con"rmwhether there is a maximum speed which givesoptimal alignment. For our choice of w

iand time

step, the critical speed sw+132 km hr~1 which is

out of the range of biologically realistic speeds.Fibroblasts move anywhere from 12 to60 kmhr~1 in three-dimensional collagen gels(Friedl et al., 1998). For some speed ranges, the

FIG. 6. Collagen alignment patterns for di!erent cell speeds. As the cell speed increases the alignment also increases. Thespeed varies from 5 kmhr~1 in (a), 15 kmhr~1 in (b) to 60 kmhr~1 in (c). Other than the speed the set-up and parameters arethe same as Fig. 3.


simulations are not sensitive to small changes inspeed. With the parameter sets tested, we "ndthat for speeds ranging from 15 to 60 kmhr~1,small changes in speed have little e!ect on theoverall amount of alignment.

3.3. CELL POLARIZATION: o

Fibroblasts moving on a "brous substratumtend to become highly elongated with a clearleading edge in what is called a bipolar form(Grinnell & Minter, 1978), as opposed to havinga more spread morphology with a large, non-localized, leading lamella when placed ona smooth surface (Grinnell & Bennett, 1981). Asstated previously, the model cells a!ect thematrix in the support of w

i, which realistically

should more closely match the area covered bythe cell membrane. Yet having a greatly elon-gated support for w

inot only complicates the

model but is also an oversimpli"cation of thecollagen production and orientation processes.Production occurs in involutions within the trail-ing edge of the cell and not throughout its entire

length (Birk & Trelstad, 1986). Thus, we takea square support for w

iand account for the elon-

gation of "broblasts by allowing the model cellsto be polarized, which means they have a prefer-ential direction; the degree of polarization isindicated in our model by the parameter o [seeeqn (2)]. As is evident by Fig. 7, this parametergreatly in#uences the path and "nal position ofthe cells. When o"0 the cell turns back on itselfand its "nal position is close to the initial posi-tion. As o is increased the cell path becomessmoother and the "nal location of the cell isfarther from its initial position, but the mostsigni"cant e!ect is during the transition in ofrom zero to non-zero. Even for relatively highvalues of o, such as 0.9, the cell deviates signi"-cantly from a straight course, since at each timestep there is a slight alteration in direction,and the cumulative e!ect of this rapidly becomessigni"cant.

It is clear from the cell paths that includinga non-zero o is more realistic, since in practicecell types which tend to polarize do not take suchcircuitous paths (Friedl et al., 1998).


Interestingly, although the cell paths are verysensitive to changes from zero to non-zero valuesof o, the collagen alignment pattern is not. In fact,as o increases from zero the degree of alignment

FIG. 8. The alignment patterns for simulations which di!eralignment is not sensitive to the polarization parameter until itThe parameters and set-up are the same as those for Fig. 3.

FIG. 7. A cell path is shown for several simulations whereonly the cell polarization parameter o is varied. The cell pathstraightens considerably once o is non-zero. The varioussymbols show the cell's location at various times duringa simulation. The data shown represents a run of about33 min (k"0.045 hr.)

changes slightly and only as o approaches 1 doesthe e!ect on the alignment become signi"cant(see Fig. 8).

3.4. INITIAL FIBRE ORIENTATION

An initial aligned region can determine theoverall alignment of the domain. We tested thein#uence of aligned regions by using initialmatrix orientations which included such regionsbut otherwise have the same set-up as the pre-vious simulations. In the standard set-up al-though the cells are uniformly distributed in thedomain, they are polarized in randomly chosendirections. The initial collagen gel is randomlyoriented with a strip running vertically throughthe domain which has a horizontal orientation. Ifthe width of this strip is su$ciently large it willdetermine the overall alignment of the regionregardless of cell polarization. As the width isdecreased the e!ect on the alignment also de-creases until for su$ciently small strips of align-ment there is no e!ect. Cell polarization changesthe rate at which the overall alignment decreasesas the width of the strip decreases, with the rate

only in the amount of cell polarization assumed. Collagenapproaches 1: (a) o"0, (b) o"0.1, (c) o"0.7 and (d) o"0.9.

FIG. 9. The e!ect of having an initial aligned region insimulations with and without cell polarization. The align-ment patterns are shown for simulations with an initial30 lm strip located approximately halfway in the horizontaldirection and running throughout the domain verticallywhere the collagen is aligned horizontally. The amount ofpolarization is the only di!erence between (a) where o"0and (b) where o"0.7. Other than the initial orientation ando, the set-up is the same as that for Fig. 3.


being smaller with higher cell polarization. Anexample of this is shown in Fig. 9.

In a particular set of simulations a strip ofalignment 20 km wide did not produce any over-all alignment e!ect with either o"0 or o"0.7.As the width of the strip was increased to 50 km,the case with no cell polarization showed onlyslightly more horizontal alignment than thenormal case, whereas when o"0.7 the horizon-tal alignment dominated the "nal "bre con"gura-tion. A strip of 100 km in width was su$cientlylarge to align the overall "nal pattern.

These results are also in#uenced by the abilityof cells to reorient collagen "bres, as representa-ted by i. In some cases which result in globalalignment, an increase in the value of i alters theresult so there is less or no alignment. This is dueto the same e!ect shown in Fig. 4, but here thecells more e!ectively realign and thus randomizethe initially aligned strip. One possible implica-

tion of the in#uence of aligned strips is that verythin scars will diminish with time while the"broblasts remain active, whereas large scars willbecome more severe. Thus if the "broblasts ini-tially set up a small region of alignment, in themonths after wounding as they continue to re-model the tissue the alignment should diminishcausing less scarring, whereas if the initial alignedregion is large enough the remodelling of the"broblasts will enhance the alignment and causemore severe scarring. This may be a factor inwound healing abnormalities such as keloidscars. In addition, due to the in#uence of i on thisprocess it may be possible to devise experimentswhich would determine this parameter by seeingif initially aligned regions grow or diminish.

3.5. CELL DENSITY

As expected, an increase in cell densityincreases the alignment of the collagen matrix.Biologically this e!ect should be enhanced since"broblasts exhibit contact inhibition. In otherwords, when two cells touch each other theyretract and go in altered directions. At low dens-ities this will not change our results, but it hasbeen shown that for high cell densities contactinhibition alone can cause cells to align in paral-lel arrays due to direct interaction with neigh-bouring cells (Mogilner & Edelstein-Keshet,1996). This would certainly enhance the align-ment of the collagen gel, but is an aspect whichwe do not include in our model since the densityof "broblasts in the dermis is relatively sparsewhen compared to those used in the in vitroalignment studies (Erickson, 1978).

3.6. CELL FLUX

The manner in which cells are introduced intothe domain plays an important part in determin-ing the alignment of the collagen gel. When thecells enter from the boundary they all havea component of motion in the same direction andtend to reinforce that component leading to morealignment. In the simulations presented thus far,the cells have been initially placed uniformly inthe gel. This eliminates the aforementioned align-ment e!ect and also gives a lower local densityfor the cells. The simulations in Fig. 10 illustratecases where the cells enter the domain from the

FIG. 10. The alignment patterns when cells enter the do-main from the boundaries. In (a) o"0.9 and the cells enteruniformly along all the boundaries and in (b) o"0 and thecells enter only at the base. In both cases the cells' previousdirection is randomly set to be a direction in the half planepointing into the domain. All other parameters as in Fig. 3.


boundaries. When the cells are not polarized,o"0, the e!ect on alignment is small, but withpolarization included, the manner in which thecells enter the domain strongly in#uences theoverall alignment patterns.

3.7. BOUNDARY CONDITIONS

With the current domain size and cell speed,the boundary conditions do not signi"cantly af-fect the results during the time relevant for ourapplications. However, results of the simulationsafter long times can be strongly in#uenced by theboundary conditions because the number oftimes a cell interacts with the boundary increaseswith cell speed. We have limited our runs to100 hr since this is the time of greatest biologicalsigni"cance; moreover, experiments dealing with"broblasts moving in gels typically run on thistime-scale. In wound healing, experimental datais typically collected up to 21 days post-wound-ing, but it is believed that the important factorsdetermining alignment occur soon after wound-

ing and so we only consider the "rst six dayspost-wounding. Fibroblasts typically begin en-tering the wound region within 24}48 hr post-wounding leaving the remaining 4}5 days as therelevant time.

4. The Tissue Regeneration Model

Having acquired a basic understanding of thematrix orientation model, we now developa more complicated model by adding featuressuch as matrix production. In order to motivatethe additional features, we begin by discussingsome details of the wound-healing process.

When the skin is wounded, a sequence of com-plex and overlapping processes are initiatedwhich result in the repair of the wound. One ofthe "rst events is the polymerization of the pro-tein "brin into a "brous mesh forming the bloodclot (Clark, 1996). This temporary extracellularmatrix has several functions including the provis-ion of a framework or sca!olding which the"broblasts and other cells can use to move intothe region. Fibroblasts typically enter the woundregion between 24 and 48 hr after wounding. Atthis time they begin to replace the "brin clot withan extracellular matrix composed of di!erentproteins, until eventually they construct a matrixprimarily of collagen which restores the integrityof the skin.

In order to incorporate tissue regeneration thematrix orientation model is modi"ed in threefundamental ways. First, the extracellular matrixis considered to be composed of two "brousnetworks, namely collagen and "brin. In this waythe domain can simulate normal tissue composedof collagen, a blood clot composed of "brin orsome combination of the two. Second, we extendthe model to include the ability of "broblasts toalter the composition of the matrix by producingand degrading the proteins. Finally, the speed ofthe "broblasts is determined by the compositionof the extracellular matrix. It is well known that"broblasts move at di!erent speeds on di!erent"brous substrates. We retain the assumptionsfrom the previous model that the "broblasts alterthe matrix alignment and obtain contact guid-ance cues from the matrix.

In this amended model the "broblasts areagain discrete objects whose paths are given by

FIG. 11. The graph of the function showing the depend-ency of the cells speed on the collagen density. The collagendensity is scaled so that it remain between 0 and 1.5.


f i(t). The extracellular matrix is representated bytwo vector "elds, c (x, t) representating the col-lagen network and b (x, t) representating the "b-rin network or blood clot. As before, the directionof the vector denotes the predominant directionof the "bres at the speci"ed point in the plane,and the vector representation is cell dependent,satisfying eqn (6) for each vector "eld c and b.Unlike the previous model the vectors do nothave unit length; rather, the length of the vectorsrepresents the density of the protein which isaltered by the "broblasts.

The "broblasts receive directional cues andspeed information from both components of theextracellular matrix. Thus eqns (1) and (2) aremodi"ed in the following manner:

f0 i (t)"s (DD c DD, DD b DD)v (t)

DDv (t)DD(7)

with

v (t)"(1!o)u (f i(t), t)

DDu (f i(t), t)DD#o

f0 i(t!q)

DDf0 i(t!q)DD(8)

and

u (x, t)"(1!c) c (x, t)#cb (x, t). (9)

As before, f0 represents the time derivative of f, o isa positive constant representing the degree towhich the cells are polarized, q is a time lag,s represents the speed (no longer constant) andc is a positive parameter which determines howmuch in#uence the "brin network has on thecell's direction. Blood clots have a high concen-tration of "bronectin, an adhesive protein onwhich "broblasts migrate more readily than oncollagen (Grinnell & Bennett, 1981). We assumethat the "bronectin is uniformly distributedthroughout the blood clot and is proportional tothe density of "brin. Thus, the speed of the "bro-blasts depends directly on the collagen and "brindensity, although it is really the "bronectin andnot the "brin which is important. The speedfunction is assumed to be the product of twofunctions, one which depends on the collagendensity and is decreasing and another which de-pends on the "brin density and is increasing. The

functional form of the collagen dependence isshown in Fig. 11 with the important feature ofbeing relatively insensitive to low collagen densit-ies. The "brin dependence is taken to be linear forsimplicity.

Overall the speed can range from a maximumvalue, denoted by l, to a minimum of about l/9(for more details see the appendix).

When "broblasts migrate into the wound re-gion they remove the blood clot by degrading the"brin and eventually replacing it with a collagennetwork, via both production and degradation ofthe extracellular matrix (Je!rey, 1992). The rateat which they produce collagen is highly depen-dent on both the composition of the extracellularmatrix surrounding them and their chemical en-vironment (Clark et al., 1995; Tuan et al., 1996).For simplicity, we model the changes in collagenand "brin density with the following ordinarydi!erential equations:

d DDc (x, t) DD

dt"(p

c!d

cDD c (x, t) DD)

N+

i/1

wi(x, t), (10)

d DDb (x, t) DD

dt"!d

bDD b (x, t) DD

N+

i/1

wi(x, t), (11)

where wi

is de"ned in eqn (4), N is the totalnumber of cells, p

c, d

c, and d

bare all positive

constants. The "rst term on the right-hand sideof eqn (10) models collagen production at aconstant rate by each cell, and the second term


models degradation at a rate proportional to thedensity of collagen already present. The contribu-tion of a cell to the collagen modi"cation is givenby multiplying the two terms by w

i, then each cell's

contribution is added up to give the total produc-tion rate. As before w

iinterpolates the in#uence of

the discrete cells to the nearby continuum "brevariables. Equation (11) for the evolution of the"brin density is similar to eqn (10) for the col-lagen density evolution but the cells do not pro-duce "brin, so there is no production term.

The extracellular matrix is reoriented in thesame manner as before and is described by eqns(5) and (3). This completes the description of thetissue regeneration model.

5. Results of the Tissue Regeneration Model

In the following simulations our standard set-up has initial conditions where at any point thereis only one matrix type. Thus, the domain isdivided into non-overlapping regions with eithercollagen or "brin networks. The "brin is random-

FIG. 12. Results from a typical simulation where the cells arorientation and density are shown. The lines representing the cogrey scaled to represent the density with white representing no cthe "brin density is shown with black representing "brin densityare shown with the area in black corresponding to the the su3.37 kmhr~1 and the average collagen density at t"96 isl"15 kmhr~1, c"0.5, i"5, p

c"0.64, d

c"0.44, d

f"0.6, a

ly oriented, representating a blood clot, and isdevoid of "broblasts which enter from the do-main edges. There are two basic cases: one whichcan be used to compare with the results of theprevious model and one which is the standardset-up for this amended model. The "rst basiccase has an initial randomly oriented "brinmatrix of uniform density in the domain. Thecells are initially uniformly placed within theblood clot (not relevant to wound healing) andthe results are shown in Fig. 12.

One can see that the collagen density is fairlyuniform in its distribution and the "brin has beene!ectively removed. By comparing the resultswith those in Fig. 8(d) it is obvious that there ismuch less alignment with the new model. This isprimarily due to the changes in the cell speed,which in this case gives an average cell speed of3.37, and can be demonstrated by "xing the cellspeed at 15 kmhr~1 and comparing the results(see Fig. 13).

The second base case is the same as the "rstexcept that the cells enter the domain from the

e initially uniformly placed in the domain. In (a) the collagenllagen orientation are streamlines for the vector "eld and areollagen and black representing a collagen density of 1.5. In (b)of 1.0 and white representing no "brin. In (c) the cell positionspport of the weight functions w

i. The average cell speed is

1.30. The time shown is t"100 hr, k"0.15 hr, N"600,nd o"0.9.

FIG. 13. The collagen alignment when the cell speed is"xed. This simulation is the same as that shown in Fig. 12except the cell speed is "xed at 15 kmhr~1 and the averagecollagen density at t"96 is 1.44.


boundaries with the results shown in Fig. 14. Inthis more biologically realistic case, the e!ects ofthe cells entering from the boundary on the ex-tracellular matrix can be clearly seen at theearlier time.

5.1. ALTERATIONS IN COLLAGEN PRODUCTION

AND FIBRIN DEGRADATION: pc, d

cAND d

b

As the collagen production is increased theaverage collagen density increases and the "brindensity is una!ected. This is demonstrated inFig. 15(a) and (c), which show plots of the col-lagen and "brin densities with respect to theparameter p

c. Figure 15(b) and (d) show time

plots of the average collagen and "brin proteindensities, respectively. Each line corresponds tothe time plot for one simulation. Thus, the timeplots for several simulations with di!erent valuesof p

care plotted on the same graph. In this way

the range of densities obtained as the parametersare changed is more easily seen and surprisinglythere are signi"cant #uctuations. Likewise whendbis increased the "brin density decreases and the

collagen density remains roughly the same (seeFig. 16) but again with #uctuations. These #uctu-ations are explained by the discrete nature of the

cells. When parameter values are changed, thecells traverse di!erent paths giving solutionswhich are pointwise very di!erent in space.Figure 17 shows the average collagen density forseveral simulations which di!er only in the ran-dom initial conditions used for the matrix ori-entation. The #uctuations in collagen densities inFigs 16 and 17 as well as in the "brin density inFig. 15 are of the same magnitude con"rmingthat they are independent of the parameter beingaltered.

The other e!ects of altering these parametersare predominantly due to changes in the cellspeed and can also be obtained by altering thefunctional dependence of the speed on the densit-ies. This includes the degree the cells are able toinvade the blood clot which is discussed in thenext section.

5.2. INVASIVENESS

The di!erent set-up for the tissue regenerationmodel introduces the additional feature of invas-iveness which needs to be considered. Since theblood clot is initially devoid of "broblasts, thedegree to which the "broblasts in the collagenmatrix are able to move into the "brin becomesan important issue. There are three factors whichplay a role in determining the invasiveness of thecells: cell polarization, cell speed and protein pro-duction. The protein production plays a partsince it alters the cell's speed by altering thedensities. By changing either the speed functionor the production and degradation parametersany degree of invasiveness can be obtained. Cellpolarization plays a role by determining the fre-quency with which a cell turns around ratherthan penetrating into a region.

5.3. INFLUENCE OF THE CLOT ON CELL DIRECTION: c

The "brin clot has a random structure createdby the polymerization of the "brin as the bloodplasma "lls the wound. The directional cues ofthe "brin clot have a randomizing e!ect on thedirection of the cells. In eqn (9) the in#uence ofthe "brin on "broblast direction is determined bythe parameter c. When c is small, so that the"brin does not contribute much to the cell direc-tion, there is more alignment of the collagen"bres and as c increases the collagen becomes

FIG. 14. Results from a typical simulation where the cells enter "brin matrix from the domain edges. In (a), (b), and (c) thecollagen orientation and density, the "brin density and the cell positions, respectively, are shown at t"20 hr. In (d), (e) and (f )the collagen orientation and density, the "brin density and the cell positions, respectively, are shown at t"100 hr. Comparewith Fig. 10(a). The parameters are the same as those shown in Fig. 12. The average cell speed is 3.24 kmhr~1 and the averagecollagen density at t"96 is 1.25.


more random. An example of this can be seen inFig. 18 where the value of c is varied for twosimulations. Even when c is small the randomiz-ing in#uence will, with time, have a signi"cantimpact on the cells' directions and will conse-quently alter the overall collagen alignment. Thisindicates that either the mechanisms creatingalignment in scar tissue are strong or the "brin

clot plays a minor role in directing the "bro-blasts.

5.4. INTERFACE

The di!erent way in which the cells interactwith the two types of protein "bres causes di!er-ent alignment properties. Most signi"cantly, the

FIG. 15. Graphs of the average protein densities as the collagen production is varied. It is seen that the collagen densityincreases as the collagen production increases and the "brin density remains relatively unchanged. In (a) and (c) the averagecollagen and "brin densities are plotted with respect to the collagen production rate respectively at several di!erent times. Thecollagen density is scaled to range from 0 to 1.5 and "brin density of 1 represents a typical clot density. The time plots (b) and(d) show the evolution of the collagen and "brin densities respectively with each line representing a di!erent simulation as thecollagen production rate is changed. Other than varying p

cand d

cwhile keeping their ratio a constant 1.5, the parameters and

set-up is the same as that for Fig. 14. The values of pcused in the simulations start at 0.1 and are incremented by about 0.05 up

to 0.85.


collagen "bres direct the cells and are reorientedby them, whereas the "brin "bres are not re-oriented. This causes more alignment when thecells are in collageneous regions of the domainand the randomizing in#uence of the "brin is notreduced as the cells #ow through those regions.This is demonstrated by simulations in which thedomain is initially divided into a random col-lagen gel and a random "brin gel, with a distinctinterface between the two in the vertical direc-tion. Figure 19 shows the collagen alignmentpatterns for such a simulation with 300 cellsinitially positioned at the collagen}"brin inter-face. There is more alignment in the collagenregion with the cells being guided into alignedstreams, whereas on the "brin side a less orientedcollagen matrix is produced. By considering onlythe speed e!ect, one would expect opposite re-sults since the cells are moving more slowly in the

collagen region than in the "brin region. By elim-inating the speed dependence and making thecells travel at a constant speed, which is close tothe average speed in the previous simulation,similar results are obtained [Fig. 19(b)]. Thisdemonstrates that speed is not the control-ling factor. However, if the speed is a constant15 kmhr~1 strong alignment is attained onboth sides showing how di!erent e!ects combineto in#uence the outcome. The higher speedtends to increase the alignment and, for thisexample, the e!ect of the high speed overcamethe randomizing in#uence of the "brin clot.At the lower speed the randomizing in#uence ofthe "brin clot determines the collagen alignmenton the "brin side, but on the other region thealignment is due to the mechanism relating tohow the cells interact with the "bres and not thespeed.

FIG. 16. Graphs of the average protein densities as the "brin degradation is varied. The "brin density decreases as thedegradation parameter increases and the collagen density remains relatively unchanged. In (a) and (c) the average collagenand "brin densities are plotted with respect to the "brin degradation rate respectively at several di!erent times. The collagendensity is scale to range from 0 to 1.5 and "brin density of 1 represents a typical clot density. The time plots (b) and (d) showthe evolution of the collagen and "brin densities respectively with each line representing a di!erent simulation as the "brindegradation rate is changed. As before other than varying d

bthe parameters and set-up are the same as Fig. 14. The values of

dbused in the simulations start at 0.1 and are incremented by about 0.05 up to 0.85.

FIG. 17. The e!ect of changing the random initial orientations on the average collagen density. In (a) the collagen density isplotted with respect to the seed for the random number generator at several di!erent times. The collagen density is scaled torange from 0 to 1.5. In (b) time plots of the collagen density in simulations using di!erent seeds for the random numbergenerator determining the initial "brin orientation are superimposed. The parameters and set-up are the same as Fig. 14 otherthan the particular random initial conditions. Fifteen di!erent seeds were used.


FIG. 18. Changes in how the blood clot in#uences celldirection are shown in these collagen plots. When the "brinmatrix has less in#uence on the cell direction (a) the collagenis more aligned and when the "brin has more in#uence (b)the collagen is less aligned. Compare (a) and (b) with Fig. 14where c"0.1, 0.9 and 0.5, respectively. The average cellspeed in (a) is 3.24 and the average collagen density is 1.23.In (b) the average cell speed is 3.25 and the average collagendensity 1.13.

FIG. 19. Alignment patterns showing the e!ect of an in-itial transition from collagen to "brin. The alignment isstronger on the left side of the domain where the initialprotein was collagen and weaker on the right side where theinitial protein was "brin. The position of the initial interfacebetween collagen and "brin is indicated by the arrow. In (a)the cell speed depends on the protein densities with anaverage speed of 3.11 lmhr~1, l"15 and the total collagendensity is 1.14. In (b) the cell speed is constant and set at5 lmhr~1. The remaining parameters are the same as thosegiven in Fig. 12 with 300 cells initially placed along thecollagen}"brin interface and polarized in randomly chosendirections.


6. Discussion

We have developed and studied a matrix ori-entation model which includes the interactions ofcells being directed by their substrate and simul-taneously reorienting it. Furthermore, we haveextended this model to allow the cells to alter thecomposition as well as the orientation of thematrix. Simulations of these models have impor-tant implications concerning the mechanismswhich cause alignment. The simple model clearlyshows that cell speed, #ux, polarization, density,initial matrix orientation and the in#uence ofcells on the matrix can all have a signi"cantimpact on the overall alignment of the collagen.As was indicated by both models, depending onthe circumstances, some of these factors will con-tribute more than others in producing alignment.Most, if not all, of these can be tested experi-mentally, but the complicated interactions meanthat the result require careful interpretation; alsothe e!ects of gel contraction should be minimizedas far as possible in experimental tests. We sug-gest four types of possible experiments:

f Altering the speed of ,broblasts. Our modelpredicts that increasing the speed of the cellsshould cause greater alignment. This could bedone either with a chemoattractant (Knappet al., 1999) or by altering the integrin expres-sion levels of the "broblasts (Palecek et al.,1997). By choosing a chemoattractant such asepidermal growth factor which can increasethe speed of the "broblasts up to three-folddepending on the substratum (Ware et al.,1998) the collagen alignment should beincreased. Of course this chemoattractant alsodecreases the directional persistence of cellsmodelled here as polarization. These twoproperties of the cells have opposite e!ects onthe collagen alignment, but our models pre-dicts that the e!ect of the increased speed out-weighs the e!ect of the decreased cell polariza-tion. For example, increasing the cell speed bya factor of three and halving the polarization

FIG. 20. Simulations designed to show the e!ect of aninitial region of collagen with di!erent orientation. Initially,all the collagen is oriented vertically except a 200]200 lm2in the centre, which is oriented horizontally. In (a) i"0.5and the region of horizontally oriented collagen remainslargely unchanged after 100 hr of simulation; whereas, in (b)i"5.0 and the region with horizontal orientation haschanged shape dramatically. In these simulations o"0.7and the remaining parameters are the same as those given inFig. 3.


parameter results in a signi"cantly more alig-ned collagen matrix.

f Reducing the contact guidance of the cell. Thiscan be accomplished by inhibiting the forma-tion of microtubules with colcemid (Oakleyet al., 1997). When this is done the cells are stillmobile but do not align along very narrowgrooves in the substratum. Alternately, treat-ment with colchicine causes a rounder mor-phology of "broblasts in a "brous gel but doesnot signi"cantly alter the collagen production(Unemori & Werb, 1986). A "nal possibilitywould be to alter the substrate by using mono-meric collagen (Mercier et al., 1996). On thistype of substrate the "broblasts take a morerounded morphology and presumably haveless contact guidance. All of these interven-tions alter not only the contact guidance butalso the degree of cell polarization. Our modelsuggests both of these act to reduce alignment.Reducing cell polarization is modelled byreducing o, and although in our model weassume that the cells align exactly in thepredominant direction of the collagen, reduc-ing i should e!ectively model a reduction incontact guidance.

f E+ects of initial collagen orientation. This couldbe tested using the approach of Matsumoto etal. (1998), who transplanted pieces of tendonwhere the "broblasts had been killed into nor-mal tendon. The manner in which the structureof the transplanted tendon was altered by theinvading cells was examined in two cases:one where the transplant was the same sizeas the defect into which it was transplantedand another where the transplant was largerand therefore lax. Altering this procedureby rotating the grafted tendon, one can setup initial conditions where the collagen hasregions with di!ering orientations. The resultof two simulations with this type of initialconditions are shown in Fig. 20. The initialcollagen gel is oriented vertically with asquare patch in the centre oriented horizon-tally. The di!erent values of i show very di!er-ent results. A similar idea would be to placepieces of oriented gel (Guido & Tranquillo,1993) in randomly oriented gel. By changingthe size of the di!erently oriented regions themagnitude of i could be estimated, helping to

determine how e!ectively the cells can reorientcollagen.

f Determining the e+ects of -ux. There are manyexperiments which could be devised to deter-mine the importance of the way in which thecells enter the region. Gels can be preparedvoid of cells and with cells distributed through-out. Our simulations suggest that gels whichhave a continuous #ux of cells from the edgeswill result in more aligned collagen near the geledges than gels which have cells distributedthroughout.

The tissue regeneration model showed that therate at which the cells alter the matrix composi-tion has little direct e!ect on the matrix align-ment. In the more complicated setting of tissueregeneration there was less alignment due to boththe reduced cell speed and the randomizing in#u-ence of the blood clot. Thus, our modelling showsthat matrix alignment is a complicated processwith many factors contributing to the overall


structure including both the inherent propertiesof the "broblasts as well as the conditions inwhich they "nd themselves. Even in our simplemodel the complexity of alignment was illus-trated.

In the context of wound healing, we concludethat cell #ux is the most signi"cant alignmentmechanism, especially when coupled with cellpolarization. Yet, the initial orientation of thecollagen should not be overlooked. As the cellsmove into a wound they may establish an initialregion of alignment when regenerating the tissuewhich is then propagated or stabilized during themonths of remodelling that occur. Thus, the in-herent properties of the "broblasts are of second-ary importance to the conditions in which thecells are placed in determining matrix alignmentfor wound healing. In addition, the alignmentmechanism must be strong to overcome ran-domizing in#uences such as the blood clot.

In a complex biological process such as woundrepair, mathematical modelling plays a valuablerole because of its ability to study speci"c parts ofthe process in isolation. This approach has beenused to study epithelial repair (Dale et al., 1994),cytokine activity in scar tissue (Dale et al., 1996),wound angiogeneis (Pettet et al., 1996; Chaplin& Byrne, 1996) and the tissue mechanics underly-ing wound contraction (Olse et al., 1995; Tracquiet al., 1995; Tranquillo & Murray, 1992). Almostall of this modelling work excludes the e!ects oftissue anisotropy, which will vary dynamicallyduring healing as a result of cell}matrix interac-tions. However, experimental evidence now sug-gests that such anisotropy plays an importantrole in scar formation (Ehrlich & Krummel, 1996;Whitby & Ferguson, 1991). The models we havedeveloped here provide a framework that enablesrealistic modelling of these phenomena, includingdynamically varying anisotropy. This is particu-lary important because of recent progress in thedevelopment of potential anti-scarring therapies(Shah et al., 1995), which act by altering thegrowth factor pro"le within the wound duringhealing. A realistic mathematical model of thescar formation process would provide a powerfultool in this development, enabling the identi"ca-tion of optimal therapeutic regimes, and ourwork provides one component of such a realisticmodel.

This research (PKM) was supported in part bythe Institute for Mathematics and its Applications(University of Minnesota) with funds provided bythe National Science Foundation. Part of this workwas supported by the London Mathematical Society,under scheme 3. JCD acknowledges support fromEPSRC grant GR/K71394.

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APPENDIX A

Here we describe the numerical implementa-tion of the models and their details. Before doingso we note that for simple con"gurations of theextracellular matrix such as oriented in one direc-tion, local realignment only causes local changeswhich do not propagate into the entire domain.That is if the initial orientation is smoothon a scale of sk where s is the cell speed and kis the time step, it remains smooth in a stablemanner such that small perturbations aresmoothed.

A.1. Protein Densities

The vector "elds c and b are discretized inspace so that c6

l,m(t)"c (lh

x, mh

y, t) and similarly

for b1l,m

where hxand h

yare the space steps in the

x- and y-direction. Equations (10) and (11) de-scribing the evolution of the protein densities aresolved at the grid locations using Euler's method.The densities are constrained to be nonnegativeand the grid used has 51]101 grid points.

A.2. Protein Fibre Directions

Equation (3) is implemented with the timelag q"0.15 hr. This represents a biologically rea-sonable lag of about 9 min. It also corresponds tothe time step k for the calculations making thetime-lagged direction readily available.

Multiplying eqn (5) by cos h, applying thechange rule and trignometric identities give

d

dt(sin h)"iDD f DD (sin a!sin a sin2 h

!cos a sin h cos h) (A.1)

and similarly multiplying eqn (5) by sin h gives

d

dt(cos h)"iDD f DD (cos a!sin a sin h cos h

!cos a cos2 h) (A.2)

By changing to Cartesian coordinates wherec;"c/DD c DD, c;"(cos h, sin h), f"DD f DD (cos a, sin a)

and f is de"ned in eqn (3), eqns (A.1) and (A.2)become

d

dt(c; )#iDD f DDSc; , fTc;"i f. (A.3)

Discretizing eqn (A.3) using a "rst-order approxi-mation to the derivative and adding terms ofO (k) and higher results in

c;l,m

((n#1)k)"d (nk)#ikf (nk)

DDd (nk)#ikf (nk)DD#O (k),

(A.4)

where c;l,m

is the discretized version of c; and

d (t)"$c;l,m

(t) (A.5)

with the sign being chosen so that

Sd, fT!d'0. (A.6)

Recall that the vector representation is cell de-pendent [see eqn (6)], but the cumulative e!ect ofall the cells is being used in eqn (A.4). Thus, therepresentation needs to be de"ned with respect tothe cumulative direction of all the contributingcells. This formulation is valid for both the matrixorientation and the tissue regeneration modelsand is used for pragmatic reasons in the numer-ical algorithm. When the collagen density is al-lowed to vary, the cells will remodel the existingcollagen to the same degree regardless of thedensity. If there is no collagen then d is taken tobe zero.

A.3. Interpolation

The interpolation from the discrete cells to thevector "eld is a simple evaluation of the weightfunction de"ned in eqn (4). Other weight func-tions have been tested including a step functionwhich is constant over the same support of theone de"ned earlier. The same qualitative resultsare obtained. The interpolator I from thediscretized vector "elds to the cell location isa tensor product interpolant using quartic


Langrangian interpolation in each direction(Ralston & Rabinowitz, 1978). It is de"ned by

I (c6 , x, y)"2+

n/~2A

2+

m/~2

c6j~m,k~n

lj~m

(x)B lk~n

(y),

(A.7)

lj(x)"

(x!xj~2

) (x!xj~1

) (x!xj`1

) (x!xj`2

)

(xj!x

j~2) (x

j!x

j~1) (x

j!x

j`1) (x

j!x

j`2),

(A.8)

where xj"( j!1)h

xand the analogous y

k"

(k!1) hy

are grid points.

A.4. Cell Paths

The paths of the "broblasts, f i (t) are computedas continuous piecewise linear curves. We de"nethe direction of the linear segment starting atf i (nk) to be Ti

nwhere n"0, 1, 22K and k is the

time step. This direction is determined by imple-menting eqns (7)}(9) as follows:

Tin`1

"

vn

DD vnDD

(A.9)

with

vn"(1!o)u

n(f i[(n#1)k])#o

Tin

DD T inDD

(A.10)

and

un(x)"(1!c)I (c6 (nk),x)#cI (b1 (nk),x) (A.11)

with similar equations for eqns (1) and (2) of thematrix orientation model. The cell takes its direc-tional cues from the matrix before remodelling it,although it is the altered density which the celluses to determine its speed.

The speed is constant in the matrix orientationmodel and for the tissue regeneration model istaken to be

sn"lA

1

8#

7

8#3DD I [ c6 (nk, f i (nk))]DD6B]A

1#2DD I [b1 (nk, f i(nk))]DD

3 B , (A.12)

where l is a positive constant denoting the max-imum speed of the cell and s

nis the speed corre-

sponding to the line segment starting at f i (nk).The speed and k determine the length of the linesegments which de"ne f i. The speed function isthe product of an increasing linear function de-pending on the "brin density and a decreasingfunction of collagen density with the form illus-trated in Fig. 11. It is relatively insensitive to lowcollagen densities and decreases the speed byabout 1/4 when the collagen density is 1 and upto about 2/3 when the collagen density is 1.5 (themaximum allowed). The dependency of the speedon the "brin is a linear function which goes from1/3 when the "brin density is 0 to 1 when the"brin density is 1. Thus the speed can range froma maximum l when DD b DD"1 and DD c DD"0 toa minimum of about l/9 when DD b DD"0 andDD c DD"1.5.

A.5. Boundary Conditions

In wounds, the cells potentially can enter orleave the region from all of the wound boundarieswith the exception of the epidermal boundary.This is clearly not the case for controlled experi-ments of cells in collagen gels. It is not under-stood how the cells respond to the boundaries inthe wound, and in controlled experiments thefocus is on the interior of the gel and not theedges. We have tried various boundary condi-tions including using a toroidal surface, absorb-ing boundaries and two types of no-#ux bound-aries. The two no-#ux boundary conditions are,one where the cells re-enter the domain at ran-domly chosen sites after leaving it. The other isused in the simulations for this paper and has thecell re#ect o! the boundary by changing the signof the component of motion perpendicular to theboundary. This boundary condition was chosenbecause it is the most realistic for the epidermalboundary in the wound, can be interpreted asa no-#ux boundary for the other wound edgeswith an equal number of cells entering and leav-ing the domain. Furthermore, it simpli"es theproblem by having the same boundary conditionfor all the edges. In addition, the simulation resultsare very similar for both no-#ux boundary condi-tions and on the time-scale of 100 hr they aresimilar for all the boundary conditions examined.


A.6. Parameter Values

Although there is a signi"cant amount of ex-perimental data dealing with "broblasts and col-lagen, establishing parameter values is very di$-cult due to the nature of the experiments. Here wejustify, to the extent possible, the parametervalues used in the model.

f In the matrix orientation model the speed ofthe "broblasts is generally taken to be15 kmhr~1. Fibroblasts moving in three-dimensional collagen gels move from 12 to60 kmhr~1 in vitro (Friedl et al., 1998), but"broblasts migrating through a "brin clot whileremodelling the composition of the extracellu-lar matrix certainly move with speeds in thelower range if not slower. In the tissue regenera-tion model the maximum cell speed, l is takento be 15 kmhr~1 with typical average cellspeeds over 3 kmhr~1. The e!ects of varyingthis parameter are discussed in Section 3.2.

f Total cell number N is taken to be 600. Thisnumber gives the maximum "broblast densityin a wound region (Adams, 1997).

f The cell polarization parameter o is discussedin Section 3.3.

f The parameter representating the in#uence ofthe "broblasts on orienting the collagen is i.For i"1, a cell would reorient the collagen atits location to its direction within 4}5 hr, ascan be demonstrated by solving eqn (5) withDD f DD"1. Detailed estimates of this parameter

are unavailable, and e!ects of this parameterare discussed in Section 3.1.

f The extent to which the "brin clot guides the"broblasts is governed by the parameter c. Wetake c"0.5 assuming the contact guidancein#uence on the "broblasts to be equal forthe collagen and "brin, although numericalsimulations indicate that only near theextreme values of c"0 and 1 are the resultsof the simulation signi"cantly a!ected (seeSection 5.3).

f The time lag q can be thought of as the timedi!erence from which the cell senses the con-tact guidance cues to the time it takes to re-orient while doing the remodelling. It is takento be 0.15 hr (9 min) which means that the cellmoves at the most 2.25 km in this time interval.

f The collagen production and degradationrates, p

cand d

cas well as the "brin degradation

rate, db

are not easy quantities to determine.The experiments which measure the collagenproduction of "broblasts in wounds collectdata several days apart. Thus with the range oferrors for the measurements and only one datapoint other than zero collagen density at thetime of wounding, there was little use in tryingto "t the data. The e!ect of altering the para-meters is shown in Section 5.1. There thevalues range from 0.1 to 0.9 which in the spa-tially homogeneous case give transition timesfrom the initial conditions to steady stateranging between 5 and 60 hr.

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Mathematical and Computer Sciences - Heriot-Watt Universityjas/paperpdfs/dallon3.pdf · 1999. 9. 17. · Mathematical and Computer Sciences - Heriot-Watt University