A differential equation model of collagen accumulation in a healing wound.

Abstract

Wound healing is a complex biological process which involves many cell types and biochemical signals and which progresses through multiple, overlapping phases. In this manuscript, we develop a model of collagen accumulation as a marker of wound healing. The mathematical model is a system of ordinary differential equations which tracks fibroblasts, collagen, inflammation and pathogens. The model was validated by comparison to the normal time course of wound healing where appropriate activity for the inflammatory, proliferative and remodeling phases was recorded. Further validation was made by comparison to collagen accumulation experiments by Madden and Peacock (Ann. Surg. 174(3):511-520, 1971). The model was then used to investigate the impact of local oxygen levels on wound healing. Finally, we present a comparison of two wound healing therapies, antibiotics and increased fibroblast proliferation. This model is a step in developing a comprehensive model of wound healing which can be used to develop and test new therapeutic treatments.

Full text

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Bull Math Biol
DOI 10.1007/s11538-012-9751-z
ORIGINAL ARTICLE
A Differential Equation Model of Collagen
Accumulation in a Healing Wound
Rebecca A. Segal ·Robert F. Diegelmann ·
Kevin R. Ward ·Angela Reynolds
Received: 31 March 2011 / Accepted: 2 July 2012
© Society for Mathematical Biology 2012
Abstract Wound healing is a complex biological process which involves many cell
types and biochemical signals and which progresses through multiple, overlapping
phases. In this manuscript, we develop a model of collagen accumulation as a marker
of wound healing. The mathematical model is a system of ordinary differential equa-
tions which tracks fibroblasts, collagen, inflammation and pathogens. The model was
validated by comparison to the normal time course of wound healing where appropri-
ate activity for the inflammatory, proliferative and remodeling phases was recorded.
Further validation was made by comparison to collagen accumulation experiments
by Madden and Peacock (Ann. Surg. 174(3):511–520, 1971). The model was then
used to investigate the impact of local oxygen levels on wound healing. Finally, we
present a comparison of two wound healing therapies, antibiotics and increased fi-
broblast proliferation. This model is a step in developing a comprehensive model of
wound healing which can be used to develop and test new therapeutic treatments.
R.A. Segal (?) · A. Reynolds
Department of Mathematics, Virginia Commonwealth University, Richmond, VA 23284-2014, USA
e-mail: rasegal@vcu.edu
R.A. Segal · R.F. Diegelmann
Center for the Study of Biological Complexity, Virginia Commonwealth University, Richmond, VA
23284-2030, USA
R.F. Diegelmann
Department of Biochemistry & Molecular Biology, Virginia Commonwealth University Medical
Center, Richmond, VA 23298-0614, USA
R.F. Diegelmann · A. Reynolds
VCURES, Virginia Commonwealth University, Richmond, VA 23298, USA
K.R. Ward
Michigan Critical Injury and Illness Research Center, Department of Emergency Medicine,
University of Michigan, Ann Arbor, MI 48109, USA

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R.A. Segal et al.
Keywords Acute wound · Collagen · Mathematical modeling · Computational
modeling
1 Introduction
The care of acute and chronic wounds is a source of significant health care cost.
Diabetes, in particular, often leads to chronic wounds that require care for the life of
the patient. It has been estimated that the cost of treating chronic wounds is $5–$10
billion a year (Kuehn 2007). Additionally, patients sustaining polytrauma often have
complications from wounds, which can secondarily lead to sepsis, multi-organ failure
and death.
The process undertaken by the human body to close a wound is complex. There
are four stages to the wound healing process (Goldberg and Diegelmann 2010), each
of which is subject to derailment and is a potential cause for failure in satisfactory
wound closure. The four stages of wound healing are hemostasis, inflammation, pro-
liferation, and remodeling. The stages overlap in time and each involve many differ-
ent biochemical processes and cell types. An understanding of the interplay between
the key mediators from each stage will lead to a greater understanding of the wound
healing process.
In recent years, there has been an effort to use mathematical modeling to help un-
derstand the complexities involved in wound healing. A review of recent modeling
efforts intheareaofwoundhealing(SherrattandDallon2002) addressesthedifferent
aspects of wound healing and outlines many of the modeling techniques relevant to
the field. Because wound healing involves so many biological processes, there have
been many different types of modeling technique used. Some research is concerned
with the geometry of the wound and focused on the formation of the skin matrix
(Almeida et al. 2011; Javierre et al. 2009). These models do not implement the effect
of inflammation, an important stage of wound healing, particularly for wounds that
fail to heal. Other research has been focused on the question of tissue oxygenation
and how supplementation or reduction of oxygen impacts the wound healing pro-
cess (Friedman et al. 2010; Flegg et al. 2010). Because of the geometric nature of
angiogenesis, these models are implemented with partial differential equations and
are computationally expensive. Finally, there is research focused on the biochemical
processes involved in the regulation of the wound healing process (Menke et al. 2010;
Reynolds et al. 2006). Our model builds on the research in this area by adding further
complexities to these existing models.
2 Model Development
This model aims to capture the dynamics of healing tissue by monitoring collagen
accumulation in a wound. Collagen is the major protein component of the wound
matrix and serves as a useful marker because it can be measured during the healing
process. Additionally, changes in collagen production during the phases of wound
healing are well correlated with actual healing. Collagen is produced by fibroblast

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A Differential Equation Model of Collagen Accumulation
Fig. 1 Model schematic for collagen accumulation model where WS0is the initial wound size. The cre-
ation of a wound triggers the recruitment of fibroblast from the healthy tissue at the edge of the wound as
well as the inflammatory response, dashed arrows. The fibroblasts proliferate and migrate. Arrows repre-
sent up regulation, while bars represent down regulation and/or inhibition
cells, one of the major cell types involved in wound healing. The differential equation
model tracks changes in the amount of pathogens, inflammatory cells, collagen and
the different stages of fibroblasts. Wound size is also tracked, as a direct function of
the collagen deposition.
Figure 1 depicts the relationship between the cell types. An initial wound event
(WS0) triggers an inflammatory response. An additional inflammatory response may
be induced by the introduction of pathogens (P) into the wound. There are many
different inflammatorycell typesand mediatorswhichmay be activatedandproduced
during the wound healing process. These are not explicitly modeled, but are grouped
together and are represented with the variable N. The activated inflammatory cells,
along with their byproducts, cause further tissue damage by destroying fibroblasts
(FIB) as well as the newly formed collagen (COL).
2.1 Variables and Parameters
The model contains a large number of parameter values (Table 1). The values for
many of these parameters were taken from Reynolds et al. (2006). The units for the
model variables and associated parameters are non-specific because they represent
the combination of various cells, signaling proteins and mediators. Therefore, most
of the variables represent a relative level of response, rather than an exact cell count.
The variable COL represents a percentage of space filled by collagen and is therefore
dimensionless. The variable P represents the concentration of pathogen cells within
the wound. The remaining parameter values in the table were estimated using the
known time course for normal wound healing and then confirmed using the collagen
accumulation data (Madden and Peacock 1971).
2.2 Fibroblast Equations
Thefibroblastpopulationisdividedintothreedifferentgroupsdependingonthecell’s
primary activity. Fibroblasts are recruited to the wound site after the initial wound
insult in response to the inflammatory activity (Diegelmann and Evans 2004). These

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R.A. Segal et al.
Table 1 Parameter values for collagen accumulation model
NameDescriptionBaseline value
μfib
pfibx
df
kfnx
xfn
pbl
kcf
xcf
hfc
Decay rate of the fibroblasts
Proliferation rate of FIBx
Transition rate of the FIBx
The rate at which N destroys FIBx
The Hill constant for the destruction of fibroblasts by N
Baseline proliferation becomes nonzero when COL·FIB > 0.01
The rate at which FIBaproduces COL
The Hill constant for the production of COL by FIBa
The inhibition exponent for contact inhibition in COL production
term
The rate at which N destroys COL
The rate at which FIBadegrades COL
The Hill constant for degradation of COL by FIBa
Threshold below which N must fall before the wound can enter
the remodeling stage
Rate at which P activates N
Rate at which the immune mediators activate N
Rate at which the wound (WS) activates N
Source rate for resting inflammatory cells
Decay rate of resting inflammatory cells
Rate at which FIBadestroys N
Decay rate of N
The inhibition constant for the inhibition of FIBx
proliferation/transition by N.
The inhibition constant for contact inhibition of FIBx
proliferation by collagen (COL)
The inhibition exponent for contact inhibition of FIBx
proliferation by collagen (COL)
The inhibition constant for contact inhibition in COL production
term
The inhibition constant for contact inhibition in COL degradation
tem
The inhibition constant for the inhibition of inflammation N by
active fibroblast FIBa
Maximum of the pathogen population
Pathogen growth rate when O2is at its normal level
Maximum increase in the growth rate of pathogen possible due to
reduction in O2levels
Normal O2level
Rate at which M destroys pathogen P
Source of background immune mediators M. These are initially
present at the wound.
Decay rate of M
Rate at which M is activated by P
Rate at which inflammation N destroys P
Determines how altered O2level determine effective wound size
0.1/day
m: 0.6, p: 0.4, a: 0.3/day
0.3/day
m: 0.5, p: 0.4, a: 0.3/day
0.6 N-units
0.08 F-units/day
1/day
4.2 F-units
1
kcn
kcfr
xcfr
Ncrit
2/N-units/day
500/day
5 F-units
0.01 N-units
knp
knn
knw
snr
μnr
knf
μn
n∞
0.5/P-units/day
3/N3-units/day
2/day
2N-units/daya
2.88/daya
0.1/F-units/day
1.2/daya
0.6 N-units
c∞
0.5
hc
4
cf∞
0.8 COL-units
cfr∞
1.8 COL-units
F∞
6 F-units
P∞
kpg0
βp
20×106/cca
0.55/daya
0.3/day
Ocrit
kpm
sm
25
0.6/M-units/day
0.12/M-units/daya
μm
kmp
kpn
wsg
0.048/daya
0.108/P-units/day
0.2/N-units/day
0.6
aParameters determined from Reynolds et al. (2006)

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A Differential Equation Model of Collagen Accumulation
Fig. 2 Fibroblast stages during wound healing with and without an inflammatory response. Transients
for proliferating fibroblasts (FIBp), migrating fibroblasts (FIBm) and active fibroblasts (FIBa) with
WS0= 0.6. All other variables were initially set to background levels, COL = 0, N = 0, FIBp= 10,
FIBm= 0, and FIBa= 0. (a) Shows activity with a normal inflammatory response. (b) Shows activity
with inflammatory response suppressed
cells then progress through three stages of activity. They proliferate (FIBp), then they
migrate further into the wound (FIBm) and, once in place, they become activated and
produce collagen (FIBa). In each case, we model the relative activity of cells, not an
actual cell count.
Each stage of fibroblast activity is modeled by a separate differential equation.
However, all stages of fibroblasts have some basic dynamics in common. Fibrob-
lasts have an intrinsic death rate and a proliferation rate which are proportional to
the cell activity, −μfibFIBx and pfibxFIBx, respectively. In addition, each stage of
fibroblast is negatively impacted by the presence of inflammation. Activated inflam-
matory cells destroy fibroblast cells (Diegelmann and Evans 2004), as well as hinders
the fibroblasts’ function at a rate proportional to both the current fibroblast popula-
tion and a Hill function of N, −kfnxfH(N,xfn)FIBx. The Hill function is of the form
fH(x,V) =
flammatory cells to destroy the fibroblast has maximum rate, kfnx, for each of the
fibroblast populations.
In normal skin there is a background level of resident fibroblasts, which are avail-
able to be recruited from the surrounding tissue and migrate into the wound from its
less damaged outer margins immediately after the wounding event. These fibroblasts
exist in both pre-wounded tissue and in healed tissue. Therefore, an initial condition
of FIBp= 10 is used to represent the initial availability of proliferating fibroblasts
in the surrounding healthy tissue at the margin of the wound following the wound-
ing event. A diffusion type term is used to model the transition from one stage of
fibroblast to the next, −dfFIBx.
Figure 2 shows the relative size and time scale for the influx of the different fibrob-
last stages with and without inhibition by inflammation. In both panels FIBp= 10 at
the start of the simulation. Explicitly modeling the three stages of fibroblasts allows
for a slower onset of collagen production in the model without using a delay function.
Notice in Fig. 2a that with inflammation FIBadoes not reach ten percent of its max-
imum until approximately 2.2 days, allowing for a delay between the wound event
and the availability of FIBafor synthesizing collagen.
x
V+x. This function is used to capture the fact that the ability of the in-

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R.A. Segal et al.
The newly synthesized collagen is structurally weaker than native collagen. Fi-
broblasts continue to respond to the presence of this weaker new collagen. Activated
fibroblastsremainatthewoundsiteafterthewoundhashealed(GoldbergandDiegel-
mann2010).Therefore,theactivatedfibroblast(FIBa)concentrationremainsnonzero
even in a closed wound. To capture this dynamic, we include a source term, pbl, in
the (FIBa) equation. The pblterm becomes nonzero once significant levels of new
collagen are present. The above dynamics leads to Eqs. (1)–(3):
dFIBp
dt
dFIBm
dt
−kfnmfH(N,xfn)FIBm
dFIBa
dt
with fH(x,V) =
2.3 Collagen Equation
= −μfibFIBp+pfibpFIBp−dfFIBp−kfnpfH(N,xfn)FIBp
(1)
= −μfibFIBm+pfibmFIBm−dfFIBm+dfFIBp
(2)
= −μfibFIBa+pfibaFIBa+dfFIBm−kfnafH(N,xfn)FIBa+pbl
x
V+x.
(3)
Collagen is synthesized by activated fibroblasts. Once the fibroblasts have prolif-
erated to sufficient levels and have migrated to the appropriate location, they will
begin producing collagen fibers, which is modeled with a nonlinear Hill-type term
kcffH(FIBa,xcf).
When inflammatory cells are present, they release enzymes that can degrade the
collagen fibers, and this activity is represented by a mass action term, −kcnNCOL.
After the tissue has been injured, fibroblasts continue to produce collagen for the
life of the tissue (Schilling 1968; Ross 1968). This would lead to excessive scarring,
so in the absence of inflammation (i.e., once the wound has healed), fibroblasts also
degrade the collagen, −kcfrfH(FIBa,xcfr)sh(Ncrit− N). A smooth approximation of
the Heaviside function, sh, is used, so that this term has an effect only when N is
sufficiently low. Once inflammation drops below Ncrit, sh(Ncrit− N) will be one
and the term will have an effect on collagen accumulation. The effects of contact
inhibition, which will be discussed and included later, control this degradation term
in the early stages of healing when N is increasing from zero and has not reached
Ncrit. This dual activity of fibroblasts characterizes the stage of wound healing known
as remodeling (Goldberg and Diegelmann 2010). Combining all these dynamics we
derive the collagen equation, Eq. (4):
dCOL
= kcffH(FIBa,xcf)−kcnNCOL−kcfrfH(FIBa,xcfr)sh(Ncrit−N)
with sh(x) =
Figure 3 shows the typical time scale for entering into remodeling for a small,
uninfected wound by plotting the rate of degradation and production of COL by
FIBa. The production and degradation curves are plots of kcffH(FIBa,xcf) and
−kcfrfH(FIBa,xcfr)sh(Ncrit−N) respectively,duringnormalwoundhealing.Therate
of degradation is zero until the inflammation in the wound is eliminated. A balance
between degradation and production is quickly met once inflammation is eliminated,
but active fibroblasts never return to zero.
dt
(4)
1
1+e−50x.

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A Differential Equation Model of Collagen Accumulation
Fig. 3 Remodeling Stage. Rate
of collagen production
(kcffH(FIBa,xcf,hf)) and
collagen degradation
(−kcfrfH(FIBa,xcfr,hfr)×
sh(Ncrit−N)) by FIBaduring
wound healing with WS0= 0.6.
All other variables were initially
set to background levels,
COL = 0, N = 0, FIBp= 10,
FIBm= 0, and FIBa= 0
2.4 Inflammation Equation
The inflammatory response is complex. Previously, Reynolds et al. (2006) derived
a detailed model of an acute inflammatory response. This is the type of response
that is important in an acute wound. The inflammation equation developed be-
low builds on this previous work which is summarized and expanded upon. Rest-
ing inflammatory cells are recruited to the wound site by pathogens, damaged
tissue, and other inflammatory cells and mediators (Martin and Leibovich 2005;
Eming et al. 2000). Since N represents all activated inflammatory cells and asso-
ciated mediators collectively, a term for the activation of N from resting inflamma-
tory cells is a function of multiple variables. This activation is triggered by the size
of the wound (WS), the amount of pathogen in the wound (P) and the amount of
pro-inflammatory mediators produced by inflammatory cells (Diegelmann and Evans
2004). These mediators are not explicitly accounted for: instead, the model uses the
level of activated inflammation (N) as a measure of aggregate inflammatory mediator
levels. Therefore, activation is dependent on three variables N, WS, and P giving a
rate of activation of R = knpP + knnN3+ knwWS. An exponent of three on the pro-
inflammatory variable is used to account for the accumulation of pro-inflammatory
cytokines needed to sustain a positive feedback loop within the pro-inflammatory re-
sponse. Low levels of N will not cause significant levels of activation, but once N
accumulates it can sustain activation of inflammatory cells and the resulting produc-
tion of pro-inflammatory cytokines. The process of activation occurs on a faster time
scale than the other interactions included in this model, and this eliminates the need
to explicitly track the resting population of inflammatory cells. A quasi-steady state
assumption on the resting population is used and this simplifies the activation process
to a single term,
μnr+R. Further details on the derivation of this type of term are in
Reynolds et al. (2006). Because R is dependent on N there is a positive feedback
loop that promotes sustained inflammation. This process is controlled during wound
healing by inhibition, which is derived later.
Fibroblasts are recruited to the wound site by mediators released from the in-
flammatory cells and are then able to repair tissue damage and modulate the inflam-
matory response (Diegelmann and Evans 2004; Menke et al. 2006; Grinnell 1994;
Buckley et al. 2001; Smith et al. 1997). This modulation of the inflammatory re-
sponse is represented with the mass action term, −knfFIBaN. The inflammatory cell
population will decrease at a greater rate when fibroblasts levels are higher.
snrR

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R.A. Segal et al.
Furthermore, there is an intrinsic decay of the inflammatory cells −μnN based on
their half-life (Harley et al. 1990; Greenhalgh 1998). Combining all of these dynam-
ics the equation for inflammation is formed, Eq. (5):
dN
dt
=
snrR
μnr+R−knfFIBaN −μnN
(5)
2.5 Inhibition
Equations (1)–(5) are derived above without modeling inhibition of the cellular pro-
cesses. To more accurately model the cellular and biochemical activity, inhibition
must be included. Many of the variables included in this model inhibit each other
using various mechanisms. In general, inhibition has been grouped into two types.
The first is cell-mediated inhibition modeled by a Hill-type inhibition function, fi,
which is dependent on the cell type being inhibited, x, and cell type causing the
inhibition, V, and is of the form fi(x,V,V∞) =
tact inhibition, is modeled as a function of collagen accumulation. As collagen in-
creases, filling the wound, contact inhibition between the cells and the collagen ma-
trix down regulates wound healing activity. Again this nonlinear process is mod-
eled with a Hill-type inhibition function. This function, which has the general form,
fc(COL,c,h) =
plier in terms which are affected by contact inhibition.
Fibroblast proliferation and stage transition are both inhibited by inflammation. In
the presence of high inflammation, fibroblasts are limited in their ability to proliferate
and migrate (Martin and Leibovich 2005). Therefore, all of the proliferation and stage
transition terms in FIBxare replaced with fi(FIBx,N,n∞). It is assumed that the
level of inhibition due to inflammation is independent of the stage the fibroblast is in
and there is no current experimental data to refute this assumption.
Fibroblast activity is down regulated by the filling in of the collagen matrix due
to contact inhibition (Takai et al. 2008). Therefore, fibroblast proliferation rates will
slowinthepresenceofcollagenfibers.Asthewoundfillsin,thereislessneedfornew
fibroblasts. It is also assumed that the level of contact inhibition is independent of the
stage the fibroblast is in, so each proliferation term is multiplied by fc(COL,c∞,4).
Both the production and degradation of collagen are also subject to contact in-
hibition. Production of new collagen decreases as COL reaches 100 %, because the
wound is filling in. The production of collagen by fibroblasts is down regulated as
COL increases. This is captured mathematically by having the COL production term
multiplied by fc(COL,cf∞,1). Degradation works in the opposite manner. Thus,
when collagen levels are high, fibroblasts in the degradation term have less inhibition.
Therefore, the multiplier on the COL degradation term is (1−fc(COL,cfr∞,12)).
Fibroblasts inhibit the activation of inflammatory cells (Buckley et al. 2001). This
inhibition is accounted for by replacing the basic activation rate with an inhibited
activation rate Ri= fi(knpP + knnN3+ knwWS,FIBa,F∞). That is, when there is
high fibroblast activity, the activation of inflammation is suppressed. Once sufficient
fibroblasts have been activated and have begun producing collagen, the inflammatory
response becomes inhibited, allowing the wound to heal. F∞is determined based on
x
1+(V/V∞)2. The second type, con-
1
1+(COL/c)h, is always dependent on collagen and is used as a multi-

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A Differential Equation Model of Collagen Accumulation
the maximum value achieve by FIBain a normal healing wound, such that the timing
of the down regulation of inflammation is appropriate.
A full version of the model, with all the inhibition included in each equation, is
presented at the end of this section.
2.6 Pathogen Equation
Pathogen growth is modeled with a logistic equation, with a growth rate of kpg(O2)
and a carrying capacity of P∞. Because the anaerobic pathogen population is in-
creased in low oxygen environments (Kühne et al. 1985), the pathogen growth rate
is a function of the oxygen level in the local environment and is determined by the
function kpg(O2), Eq. (6):
?kpg0,
This relationship between tissue oxygenation levels and bacterial growth is illustrated
in Fig. 4a with Ocritset to 25, which was chosen to represent a normal transcutaneous
oxygen level of 25 mm Hg (Ballard et al. 1995; Caselli et al. 2005). When the tissue
oxygen level is above the critical value, kpg0remains fixed at 0.55.
The second term in Eq. (7) is directly from the Reynolds et al. model and accounts
for local immune mediators that immediately interact with the pathogen, such as
defensing and non-specific antibodies (Stefanini et al. 2008; Paulsen et al. 2002; Raj
and Dentino 2002).
Inflammatory cells such as neutrophils and macrophages actively seek out and
remove pathogens from the wound site. These phagocytic cells are less able to mi-
grate in a low oxygen environment (Turner et al. 1999). To account for these dynam-
ics it is assumed that the destruction term for pathogens, −kpnNP, is modified by
(1 − g(O2)) to model the dependence on O2levels. Despite high levels of inflam-
matory cells in the wound region, the hypoxic environment will impede their ability
to reach the pathogens. The function (1 − g(O2)) (Fig. 4b) is designed to capture
the decreased effectiveness of cells in a hypoxic environment and to represent bet-
ter than normal function when O2is larger than the critical value, Ocrit. In Eq. (7),
kpg(O2) =
if O2≥ Ocrit
if O2< Ocrit
kpg0+βp(1−
O2
Ocrit),
(6)
Fig. 4 Auxiliary functions for oxygen. kpg(O2) and g(O2). Plots of the functions used to implement the
effects of altering local tissue O2above or below Ocrit= 25

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R.A. Segal et al.
(1 −g(O2)) is used to modify the ability of inflammatory cells to remove pathogens
from the wound. Furthermore, the activity of the inflammatory cells is inhibited by
the presenceof activefibroblasts,so N was replacedwith fi(N,FIBa,F∞). This non-
linear inhibition of N has FIBaas the inhibitor with F∞controlling the effectiveness
of FIBaas described above. By combining the above terms, the pathogen equation is
formed, Eq. (7):
?
−kpnPfi(N,FIBa,F∞)?1−g(O2)?
where g(V) = 1−
dP
dt
= kpg(O2)P
1−
P
P∞
?
−
kpmsmP
μm+kmpP
(7)
1.1
1+0.1e−0.3(V−Ocrit).
2.7 Wound size
Wound closure is a direct result of the production of collagen fibers. These fibers
form the matrix that will lead to contraction and closure of the wound. To describe
wound closure, the model has a dimensionless variable which measures the effective
size of the wound, WS. This is a function of both COL and O2, Eq. (8). The wound
size is prevented from becoming negative by using the max function. This is nec-
essary because the variable COL is allowed to take on values above one in order to
account for scarring (due to excess collagen production and abnormal remodeling).
The effective wound size accounts for the additional tissue damage which occur in a
low oxygen environment. WS is used to determine the rate of activation for N in Ri
in Eq. (13). The wound is not considered to be larger in the low oxygen environment,
but the byproducts of tissue damage that signal activation are greater. Therefore, WS
is interpretedas theeffectivewoundsizebecauseit accountsfor theeffects of reduced
O2by including the multiplier,
?
1
1−wsgg(O2):
WS = max
(1−COL)WS0
1
1−wsgg(O2),0
?
(8)
2.8 Full Model
Combining all of the equations with inhibition the following system of equations
evolves, Eq. (9)–(15):
dFIBp
dt
= −μfibFIBp+pfibpfi(FIBp,N,n∞)fc(COL,c∞,4)
−dffi(FIBp,N,n∞)−kfnpfH(N,xfn)FIBp
= −μfibFIBm+pfibmfi(FIBm,N,n∞)fc(COL,c∞,4)
−dffi(FIBm,N,n∞)+dffi(FIBp,N,n∞)−kfnmfH(N,xfn)FIBm (10)
= −μfibFIBa+pfibafi(FIBa,N,n∞)fc(COL,c∞,4)+dffi(FIBm,N,n∞)
−kfnafH(N,xfn)FIBa+pbl
(9)
dFIBm
dt
dFIBa
dt
(11)

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A Differential Equation Model of Collagen Accumulation
dCOL
dt
= kcffH(FIBa,xcf)fc(COL,cf∞,hfc)−kcnNCOL
−kcfrfH(FIBa,xcfr)sh(Ncrit−N)?1−fc(COL,cfr∞,12)?
=
μnr+Ri
= kpg(O2)P
P∞
−kpnPfi(N,FIBa,F∞)?1−g(O2)?
WS = max
(12)
dN
dt
dP
snrRi
−knfFIBaN −μnN
?
(13)
dt
1−
P
?
−
kpmsmP
μm+kmpP
(14)
?
(1−COL)WS0
1
1−wsgg(O2),0
?
(15)
where
fH(x,V) =
x
V +x
1
sh(x) =
1+e−50x
fi(x,V,V∞) =
x
1+(V/V∞)2
fc(COL,c∞,h∞) =
?knpP +knnN3+knwWS,FIB,F∞
kpg(O2) =
kpg0+βp(1−
1
1+(COL/c∞)h∞
Ri= fi
?
?kpg0,
if O2≥ Ocrit
if O2< Ocrit
O2
Ocrit),
and
g(V) = 1−
1.1
1+0.1e−0.3(V−Ocrit)
3 Results
Using the ODE model, Eqs. (9)–(15), with the baseline parameters in Table 1, a nor-
mally healing wound is simulated. This wound was simulated using an initial wound
size of WS0= 0.6, see Fig. 5. The three panels of Fig. 5 show the transients for ac-
tivated fibroblasts (FIBa), inflammation (N) and collagen (COL). The wound filled
with new collagen on approximately the 14th day of healing and inflammation com-
pletely subsided within the next two days. Fibroblast activity is sustained after the
wound has closed as a result of the wound entering the remodeling stage of wound
healing.
When the initial wound size is increased, a chronic wound that is incapable of
healing is simulated, see Fig. 6. The initial wound size was increased to 0.8 which
resulted in an increased inflammatory response. The maximum of the inflammatory
response in the first 3 weeks in Fig. 6 (large wound) is near 0.8 whereas the maximum
inFig.5(smallwound)isaround0.5.Noticethatthefibroblastandcollagenlevelsare

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R.A. Segal et al.
Fig. 5 Transients for normal healing. This wound was created by setting WS0= 0.6 with all model vari-
ables initially set to background levels, COL = 0, N = 0, FIBp= 10, FIBm= 0, and FIBa= 0
Fig. 6 Transients for a chronic wound. This wound was created by setting WS0= 0.8 with all model
variables initially set to background levels, COL = 0, N = 0, FIBp= 10, FIBm= 0, and FIBa= 0
Fig. 7 Stages of wound healing. Transients are plotted for scaled N (red—inflammation), the sum of
FIBmand FIBa(blue—proliferation) and collagen degradation term from Eq. (12) (green—remodeling).
Each transient was scaled by the maximum of the variable model for comparison (Color figure online)
suppressed as a result of the high levels of inflammation and large initial wound size.
High levels of inflammation, separate from that produced by infection, can be the
result of the large amount of tissue destruction during the initial wounding event. In
both of the above cases, no infection was introduced so P remained zero throughout
the simulations.
Once the baseline parameters were determined based on achieving the proper time
course of healing in a moderate wound, the modeling dynamics were validated by
plotting the time course for the stages of wounding healing included in the model,
inflammation, proliferation, and remodeling. To illustrate these stages using model
variables we plotted N (inflammation), the sum of FIBmand FIBa(proliferation),
and the degradation term from the collagen equation (remodeling) in Fig. 7. The
sum of the FIBmand FIBavariables were used, because these two stages represent
the fibroblasts that are within the wound and lead to the closing of the wound. The
degradation term from the collagen equation is zero outside of the remodeling stage

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A Differential Equation Model of Collagen Accumulation
Fig. 8 Comparison of model simulations and experimental data during normal wound healing. Data from
Madden and Peacock (1971). WS0= 0.6 with pathogen, inflammation and fibroblast all initially zero. In
the left panel, data were scaled as described in the text so that they represented the same quantity as COL.
In the right panel they were scaled to represent the same quantity as the production term from Eq. (12)
of healing and so it is a good marker for when the remodeling phase has begun. FIBa
does not degrade collagen until the wound has closed. Each of these three stages was
scaledsothattheirmaximumswereone,inordertoeasilycomparetheirtimecourses.
Figure 7 shows that the initial days of wound healing is dominated by inflammation
and the initiation of fibroblast activity. Inflammation then starts to decrease, moving
the wound into the proliferation phase. In the final stage, inflammation has subsided
and the wound enters the remodeling phase. In this stage fibroblasts will degrade or
produce collagen at a rate dependent on level of the collagen.
We further validated the model using data from (Madden and Peacock 1971) in
which the amount of newly accumulated collagen and rate of accumulation in a
woundweremeasuredusing3–4H3Prolinetolabelthenewcollagen.Figure8shows
that the scaled Madden and Peacock data agree with the simulated data in the model
for an initial wound size of 0.6 (WS0= 0.6) with no pathogen present. Given that the
variable in the model, COL, does not represent the exact level of collagen present, but
rather a percentage of the wound covered, data presented in the work by Madden and
Peacock were scaled by a single multiplier. The data from Madden and Peacock are
scaled so that the end level for the collagen was slightly above one in the left panel.
Additionally, the rates of accumulation data were scaled by the day 14 data (max).
New collagen (COL) is zero in a new wound and one when the wound is healed.
Values above one can occur during the healing process and are interpreted as scaring.
Though the simulations do not match at every point, it can be seen that this model
captures the overall dynamics of the wound healing process.
Using the validated model, the healing response to wounds with nonzero initial
pathogen was simulated and is shown in Fig. 9. The dashed transients correspond
to an initial pathogen load of P0= 0.5 in a small wound (WS0= 0.6) and the solid
curves correspond to P0= 1.5 and WS0= 0.6. The higher initial pathogen level gives
rise to a strong inflammatory response that thwarts the ability of the wound to heal,
resulting in a chronic wound (COL ? 1). The lower initial pathogen level gives rise
to a larger inflammatory response than in the pathogen-free wound (Fig. 5), but the
pro-inflammatory feedback is not sufficient to sustain the inflammation. Therefore,
the wound is able to heal (COL > 1) and ultimately enters the remodeling stage.
Wounded tissue with a poor oxygen supply is known to have difficulty healing
(London and Donnelly 2000). Given this sensitivity to the oxygen level at the wound

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R.A. Segal et al.
Fig. 9 Transient solutions of wound healing with high and low initial infection levels. Solid
line—P0= 1.5 and dashed line—P0= 0.5. WS0= 0.6 and all other variables were initially set to back-
ground levels, COL = 0, N = 0, FIBp= 10, FIBm= 0, and FIBa= 0
Fig. 10 Wound response to various O2levels. All transients have WS0= 0.6 and all other variables were
initially set to background levels, COL = 0, N = 0, FIBp= 10, FIBm= 0, and FIBa= 0. The arrow
indicates the direction of increasing O2levels (10, 15, 20, 25, 30). Solid curves result in a healed wound.
The blue solid curve has O2= Ocrit= 25 and black O2= 30. The dashed curves result in a non-healed
wound, red O2= 20, black O2= 15 and blue O2= 10 (Color figure online)
site, the model was used to predict wound healing dynamics for various oxygen lev-
els. In Fig. 10, the effective wound size (WS), fibroblast (FIBa), and inflammation(N)
versus days since wounding were plotted for an initial wound size of 0.6 and with O2
levels below, above and at Ocrit. WS appears in the activation term of the inflamma-
tion equation, Eq. (13). When oxygen is reduced this term increases and results in
additional activation of the inflammatory response despite the fact that the wound
size has not actually increased. Dropping the O2levels below Ocrit= 25 (blue solid
curve) to 20 gives rise to an increase in WS and inflammation levels with decreases in
fibroblast (red-dashed curves). New collagen is able to form in the wound. However,
the wound does not fully heal; the effective wound size is nonzero at steady state. Fur-
ther decreases lead to more significant changes in the transients, as seen with O2= 15
(black-dashed) and O2= 10 (blue-dashed). Healing is nearly fully suppressed, so the
effective wound size remains very high. The fibroblasts are unable to mount an effec-
tive response and the wound is filled with inflammation. O2levels below 15.6 result
in a non-healing wound with no new collagen.
Next the model was used to explore healing when oxygen is reduced and initial
pathogen is nonzero. In Fig. 11 we present the transients for COL, FIBa, N, WS, and
P,with P0= 0.7, O2= 22, WS0= 0.6.Thiswounddoesnotheal,eventhoughthein-
fection is eliminated after 8 days. The degradation of fibroblasts due to inflammation
causes a drop in the fibroblast level. This resulting level of fibroblasts does not sustain
collagen accumulation in the presence of inflammation, which continues to degrade
the collagen resulting in a net loss. Lower collagen levels translate to larger effective

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A Differential Equation Model of Collagen Accumulation
Fig. 11 Wound healing in an infected, oxygen depleted wound. Transients for collagen (COL), fibrob-
last (FIBa), inflammation (N), effective wound size (WS) and pathogen (P) versus days of healing with
P0= 0.7, WS0= 0.6, and O2= 22. All other variables were initially set to background levels, COL = 0,
N = 0, FIBp= 10, FIBm= 0, and FIBa= 0
Fig. 12 Collagen accumulation in treated wounds with and without reduced O2. Left panel, transients
for collagen (COL) with normal healing (dot-dashed), antibiotic treatment (dashed, αt= 10), and altered
fibroblast (solid, pt= 1.3) with normal oxygen, P0= 0.7, and WS0= 0.6 and all other variables were
initially set to background levels, COL = 0, N = 0, FIBp= 10, FIBm= 0, and FIBa= 0. Right panel,
all parameters and initial conditions are the same as the left panel except O2= 20. All treatments were
implemented at t = 2 days
wound size and thus triggers further activation of inflammation around day 21, which
qualitatively captures observed inflammation dynamics in a low O2environment as
described in Eming et al. (2000).
Finally,themodelis usedtoexploretheuse of treatmentoptionsinwoundhealing.
Two treatment options were investigated: administering an antibiotic and altering fi-
broblast proliferation. A comparison of these treatment outcomes is shown in Fig. 12
in a normal wound (O2= 25, left panel) and a reduced O2level wound (O2= 20,
right panel). The transients on the left correspond to a wound with normal O2lev-
els. Antibiotic treated wounds are plotted with a dashed line, while the curves for
increased proliferation of fibroblasts are plotted with a solid line.
To implement the treatment of antibiotics, the decay of pathogens was increased
on day three by adding the term −αtP to the pathogen equation. This change in decay
was maintained for the remainder of the simulation. To alter the fibroblast prolifera-
tion, the proliferation rate of all fibroblasts was multiplied by ptthroughout the re-