Mathematical Models for Wound Healing Events
Part 1: Biological background
E. Javierre1 , F. J. Vermolen2 , P. Moreo1,3 , J. M. Garcı́a-Aznar1,3 , M. Doblaré1,3
1
Centro de Investigación Biomédica en Red en Bioingenierı́a, Biomateriales y Nanomedicina (CIBER-BBN)
Zaragoza, Spain
E-mail: [email protected]
2
Group of Numerical Analysis, Delft Institute of Applied Mathematics
Delft University of Technology, The Netherlands
3
Group of Structural Mechanics and Material Modelling
Aragón Institute of Engineering Research (I3A)
University of Zaragoza, Spain
Contents
1
Introduction
Structure and function of the skin
Wound definition and implications
2
Phases and processes during healing
Inflammatory phase
Proliferative phase
Remodelling phase
3
Types of wounds
Epidermal wound healing
Dermal wounds
Chronic wounds
Introduction
Structure and function of the skin
The skin forms the external covering of the body and is its largest organ (15-20% of
the total mass).
Layers of the skin:
- epidermis:
composed of keratinized stratified squamous
epithelium
- dermis:
composed of a dense connective tissue that imparts
mechanical support, strength, and thickness to the skin.
Functions of the skin:
- mechanical, permeability and ultraviolet barrier
- external environment sensor
- body temperature and water loss regulation
- endocrine secretion of hormones, cytokines and
growth factors
- immunologic information to appropriate effector
cells in the lymphatic tissue
Introduction
Wound: definition and implications
Wound:
(from Mediline Plus)
a a physical injury to the body consisting of a laceration or breaking of the skin or
mucous membrane
b an opening made in the skin or a membrane of the body incidental to a surgical
operation or procedure
The existance of a wound compromises organ integrity
... decreasing the mechanical strength
... putting the immunologic system at risk
... impairing organ function
Patients with impaired or unsuccessful wound healing often have a poor life quality,
aesthetic scarring and in some cases even emotional distress.
USA, 2003: 3.4 million of patients with chronic wounds ! $8000M
(Clark et al. 2007)
Phases and processes during healing
Inflammatory phase
1. Vasoconstriction and hemostasis
Ruptured cell membranes release inflammatory factors
(thromboxanes, postraglandins) which trigger the
vasoconstriction of injured blood and lymphatic vessels
preventing blood loss.
Wounding activates the coagulation cascade, triggered by
platelets-derived inflammatory factors and glycoproteins
that cause platelets to aggregate.
The blood clot serves as a provisional matrix formed of
fibrin, fibronectin, vitronectin and thrombospondin with
embedded platelets and various blood cells that provides a
scaffold for cellular migration.
From: Singer and Clark 1999
Shai and Maibach 2005
Phases and processes during healing
Inflammatory phase
2. Vasodilatation, increased permeability
Following vasoconstriction, histamine and postraglandis
induce vascular dilatation and permeability increase, which
facilitates the ingress of growth factors and white blood
cells into the wound.
The injury may become swollen due to plasma leaking into
the surrounding tissue.
3. Phagocytosis
Chemically attracted to the wound, monocytes enter the
wound through the blood vessels walls where they mature
into macrophages.
From: Singer and Clark 1999
Shai and Maibach 2005
White blood cells (neutrophils, leukocytes) and
macrophages act against pathogenic organisms, debride
the necrotic tissue and secrete other growth factors that
further activate the wound healing process.
Phases and processes during healing
Proliferative phase
1. Angiogenesis
Imperative to successful healing as supports cell function
(i.e. cellular migration and proliferation, collagen synthesis,
etc) with nutrients and oxygen.
Hypoxia stimulates macrophages and platelets to secrete
certain angiogenic factors. Endothelial cells are attracted
to the wound by fibronectin and angiogenic factors.
Endothelial cells migration and proliferation results into
sprout tips branching from existing vasculature into the
wound site. Parallel degradation of the blod clot
(fibrinolysis) is necessary.
From Singer and Clark 1999
Phases and processes during healing
Proliferative phase
2. Fibroplasia and Granulation tissue formation
Fibroblasts invasion occurs in parallel to angiogenesis. In a
first stage, fibroblasts migrate from the sides into the
blood clot adhering to fibronectin.
After a few days, fibroblasts start synthesizing collagen
(type III) to which they adhere for further migration,
which results in some sort of structural alignment that
gradually enables skin integrity to be restored.
The granulation tissue consists of inflammatory cells,
endothelial cells, collagen, fibroblasts, immature collagen
(type III) and myofibroblasts and presents a reddish and
granular appearance.
From Singer and Clark 1999
As fibroblasts proliferate and produce collagen,
proteoglycans, glycoproteins, elastin, fibronectin, the
granulation tissue is replaced with a provisional
extracellular matrix
Phases and processes during healing
Proliferative phase
3. Wound contraction
Early fibroblasts migration into the wound increases the
mechanical stress at the wound edge that in action with
cytokines induce differentiation into myofibroblasts.
Myofibroblasts are
smooth muscle
cells rich of actin
filaments that
induce the
contractile forces
on the wound edge
towards its center.
From Singer and Clark 1999
Tomasek et al. 2002
Wound contraction does not involve the formation of new
tissue but the centripetal movement of healthy tissue
surrounding the wound to achieve minimal scarring.
Phases and processes during healing
Proliferative phase
4. Re-epithelialization
Shortly after wounding, epidermal cells undergo phenotypic
alteration that includes dissolution of intracellular
desmosomes and formation of peripheral cytoplasmatic
filaments.
The free-edge effect and the absence of intracellular
adhesions induce the lateral movement of epidermal cells.
The migrating epidermal cells dissect the wound,
separating desiccated eschar from viable tissue.
Epidermal proliferation at the wound margin is required to
sustain migration. Local release of growth factors and
increased expression of growth factor receptors stimulate
epidermal migration and proliferation.
From Singer and Clark 1999
Degradation of the provisional extracellular matrix is
required to permit epidermal cells migration. This
degradation depends on the production of collagenase and
the activation of plasmin, both mediated by epidermal
cells.
Phases and processes during healing
(Scar) Remodelling phase
Continuous process of dynamic equilibrium between
the lysis of the early synthesized collagen type III
and the synthesis and subsequent alignment of
collagen type I (more stable) effectively increasing
the wound tensile strength.
Phases and processes during healing
Time perspective
Types of wounds
Classification of wounds
1
Epidermal wounds
a In adults
b In embryos
2
Full thickness or dermal wounds
3
Chronic or non-healing wounds
a Hypertrophic scars
b Keloids
c Pressure ulcers
Types of wounds
Epidermal wound healing in adults
Epidermal wounds are superficial wounds that do not affect (or barely affect) the
underlaying dermis. Example: a blister.
Epithelial cells at the wound margin undergo phenotypic alteration that gives
them the ability to move via finger-like lamellipodia:
rolling mechanism (mammals)
sliding mechanism (amphibian)
Mattila and Lappeleinen 2008
Absence of contact inhibition and the change in cell shape stimulate the mitotic
activity.
increase of the mitotic activity in a band behind the wound margin
maximal mitotic activity at the wound margin (15 x normal mitotic rate)
Types of wounds
Mathematical models of epidermal wound healing
Sherrat & Murray 1991
Model of epidermal cells migration with a mitosis-controlling chemical
“
∂n
n ”
2
= D∇ n + s(c)n 2 −
− kn
∂t
n0
∂c
2
= Dc ∇ c + f (n) − λc
∂t
activator
inhibitor
Predicted re-epithelialization in good agreement with experimental results on
circular wounds in rabbit ears
Travelling wave solutions as an estimate of the healing rate
Investigate the effect of the mitosis-controlling chemical on the wound shape
during healing
Types of wounds
Mathematical models of epidermal wound healing
Extensions to Sherratt and Murray 1991: (corneal wound healing)
Dale et al. 1994: chemical enhancement of epithelium cells diffusion
Gaffney et al. 1999: distinguish between active and quiescent cells, obtain a
better approximation of experimental mitotic rate
Adam 1999-2002:
mitogenic chemical with a discontinuous switch mechanism
∂c
= D∇2 c + P 1Ωal − λc
∂t
healing starts if and only if c ≥ θ at the wound edge
analyze conditions for healing initiation ! critical size defect
planar, circular and spherical wounds
Extensions to Adam 1999:
Vermolen et al. 2006: general wound morphologies; investigate mitogenic
concentration at the wound edge
Javierre et al. 2008: temporal evolution of the wound; study the effect of wound
morphology on the healing process; investigate conditions for incomplete healing
Types of wounds
Embryonic wound healing
Embryonic wound healing do not heal by lamellipodial crawling but instead by
circumferential tension at the wound edge.
Quick reorganization of the actin filaments of the cells at the wound margin
results in the formation of a pulling actin cable.
Experimental observations:
fast healing
scarless (perfect healing )
mechanically driven, low
concentrations of growth factors
Grasso et al. 2007
Types of wounds
Mathematical models of embryonic wound healing
Sherratt 1993: formation of the actin cable
"
`
´
∇ · G Eε + Γ∇ · uI
+
|
{z
}
elastic stress
G ≡ initial density of intracellular actin filaments
τ|{z}
GI
contraction stress
#
−
λGI
|{z} = 0
attachment to underlying substratum
Types of wounds
Dermal wound healing
A dermal wound compromises the integrity of the dermis
Lost tissue is regenerated through a sequence of partly overlapping sequential
biological processes that require highly orchestrated interactions between
cells: macrophages endothelial cells, fibroblasts, myofibroblasts, ...
chemicals: PDGF, VEGF, TGF-β, ...
proteins: collagen, fibrin, ...
Dynamic synthesis and degradation of the ECM
Types of wounds
Chronic or non-healing wounds
Fibroplasia related ! excessive connective tissue deposition
- Hypertrophic scars:
after trauma
remains within the wound confines
may regress in time
- Keloids:
genetic predisposition
extends beyond the original wound
rarely regress
require surgical intervention
Olsen et al. 1996: wound contraction model predicts formation of keloids due to
unbalanced chemical kinetics