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Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Modelling macrophage dynamicsand their therapeutic implications
in diabetic wound healing
Jonathan A. Sherratt
Department of MathematicsHeriot-Watt University
ECMTB08, Edinburgh, 1 July 2008
This talk can be downloaded from my web sitewww.ma.hw.ac.uk/∼jas
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
In collaboration withHelen Waugh
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Diabetic Wound HealingWhy is Wound Healing Impaired in Diabetics?
Outline
1 Introduction to Diabetic Wound Healing
2 A Simple Mathematical Model
3 An Expanded Model
4 Using the Model to Understand the Treatments
5 Conclusions
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Diabetic Wound HealingWhy is Wound Healing Impaired in Diabetics?
Diabetic Wound Healing
Wound healing in diabetics is impaired
In some cases, wounds fail to heal over long periods(> 6 months ↔ diabetic ulcer)
Limb amputation is required in extreme cases
Annual cost in UK is £17M
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Diabetic Wound HealingWhy is Wound Healing Impaired in Diabetics?
Why is Wound Healing Impaired in Diabetics?
There is not a clear understanding of why wound healing isimpaired in diabetics
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Diabetic Wound HealingWhy is Wound Healing Impaired in Diabetics?
Why is Wound Healing Impaired in Diabetics?
There is not a clear understanding of why wound healing isimpaired in diabetics
One component is that some aspects of repair fail toprogress beyond the inflammatory phase
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Diabetic Wound HealingWhy is Wound Healing Impaired in Diabetics?
Why is Wound Healing Impaired in Diabetics?
There is not a clear understanding of why wound healing isimpaired in diabetics
One component is that some aspects of repair fail toprogress beyond the inflammatory phase
In particular, macrophages persist for long times, withsignificant numbers present after a month (cf clearanceafter a few days in normal healing)
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Diabetic Wound HealingWhy is Wound Healing Impaired in Diabetics?
Why is Wound Healing Impaired in Diabetics?
There is not a clear understanding of why wound healing isimpaired in diabetics
One component is that some aspects of repair fail toprogress beyond the inflammatory phase
In particular, macrophages persist for long times, withsignificant numbers present after a month (cf clearanceafter a few days in normal healing)
We develop a mathematical model to investigate thispersistence of macrophage numbers (and associatedphenotype imbalance)
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Model IngredientsModel EquationsTypical Model SolutionSteady States and Model Predictions
Outline
1 Introduction to Diabetic Wound Healing
2 A Simple Mathematical Model
3 An Expanded Model
4 Using the Model to Understand the Treatments
5 Conclusions
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Model IngredientsModel EquationsTypical Model SolutionSteady States and Model Predictions
Model Ingredients
Monocytes migrate to the wound in response to TGFβ (T )
In the wound, monocytes differentiate into one of threemacrophage phenotypes, in response to micro-environmental signals:Inflammatory macrophages (φI) release various cytokines
that promote migration and proliferation offibroblasts and endothelial cells
Repair macrophages (φR) remodel the extracellular matrixCytocidal macrophages remove bacteria and other debris
(not included in the model)
The balance between inflammatory and repairmacrophages is different in normal and diabetic wounds
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Model IngredientsModel EquationsTypical Model SolutionSteady States and Model Predictions
Model Equations
dφI/dt =
influx︷ ︸︸ ︷
αK (T ) +
proliferation︷ ︸︸ ︷
k1ΦI(k2 − ΦI − ΦR)−
removal︷︸︸︷
d1ΦI
dφR/dt = (1 − α)K (T ) + k1ΦR(k2 − ΦI − ΦR) − d1ΦR
dT/dt = k3ΦI︸︷︷︸
production
− d2T︸︷︷︸
decay
αdiabetic > αnormal
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Model IngredientsModel EquationsTypical Model SolutionSteady States and Model Predictions
Typical Model Solution
Normal wound (α = 0.5):
0 20 400
125
250
Time, days
φ I, cel
ls/c
ubic
mm
0 20 400
125
250
Time, days
φ R, c
ells
/cub
ic m
m
0 20 400
3
6
Time, days
TG
F−
B, p
g/cu
bic
mm
Diabetic wound (α = 0.8):
0 20 400
500
1000
Time, days
φ I cel
ls/c
ubic
mm
0 20 400
150
300
Time, days
φ R, c
ells
/cub
ic m
m
0 20 400
4
8
Time, days
TG
F−
B, p
g/cu
bic
mm
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Model IngredientsModel EquationsTypical Model SolutionSteady States and Model Predictions
Steady States and Model Predictions
Non-negative steady states are:
Predictions:The non-healing steady state appears abruptly, via abifurcation, as α is increasedThe healed steady state persists and remains stable up toα = 1, so that effective treatment is a possibility
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Key Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft TreatmentsSimulation of Apilgraf TreatmentSimulation of Dermagraft Treatment
Outline
1 Introduction to Diabetic Wound Healing
2 A Simple Mathematical Model
3 An Expanded Model
4 Using the Model to Understand the Treatments
5 Conclusions
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Key Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft TreatmentsSimulation of Apilgraf TreatmentSimulation of Dermagraft Treatment
Key Interactions in the Expanded Model
To investigatepossibletherapies,we use anexpanded model.
� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � α 1−α
� � � � ! � ! � � � � � � � � � � � � � � � � � � � � �" � � � � � � � � � � � � � � �Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Key Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft TreatmentsSimulation of Apilgraf TreatmentSimulation of Dermagraft Treatment
Typical Solution of the Expanded Model
Model variables: inflammatory macrophages, repairmacrophages, fibroblasts, collagen, TGFβ, PDGF,hyaluronan
0 50 100 150 2000
200
400
600
800
1000
time, days
infla
mm
ator
y M
φ
Normal wound
Diabetic wound
0 50 100 150 2000
50
100
150
200
250
time, days
repa
ir M
φ
Normal wound
Diabetic wound
0 50 100 150 2000
500
1000
1500
time, days
Col
lage
n, u
g m
m−
3 Normal wound
Diabetic wound
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Key Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft TreatmentsSimulation of Apilgraf TreatmentSimulation of Dermagraft Treatment
Apligraf Treatment
Apligraf (Organogenesis, US) is an artificial skin comprisingdermal and epidermal layers in a bioabsorbable scaffold.
Component 100 % Density(per cubic mm)Neonatal Fibroblasts 1550 cellsTGF-β 0.4 pgPDGF 1 pgCollagen 0.45 µgTotal Hyaluronan 18.0 µgDermal Hyaluronan 15.5 µgEpidermal Hyaluronan 2.5 µg
Protocol: one application per week for five weeks.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Key Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft TreatmentsSimulation of Apilgraf TreatmentSimulation of Dermagraft Treatment
Apligraf Treatment
Apligraf (Organogenesis, US) is an artificial skin comprisingdermal and epidermal layers in a bioabsorbable scaffold.
Component 100 % Density(per cubic mm)Neonatal Fibroblasts 1550 cellsTGF-β 0.4 pgPDGF 1 pgCollagen 0.45 µgTotal Hyaluronan 18.0 µgDermal Hyaluronan 15.5 µgEpidermal Hyaluronan 2.5 µg
Protocol: one application per week for five weeks.
A separate modelling study suggests that the epidermal layerdoes not play a significant role in initiating wound repair.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Key Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft TreatmentsSimulation of Apilgraf TreatmentSimulation of Dermagraft Treatment
The Apligraf Web Site
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Key Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft TreatmentsSimulation of Apilgraf TreatmentSimulation of Dermagraft Treatment
Dermagraft Treatment
Dermagraft (Advanced Biohealing Inc, US) is an artificial dermallayer.
Component 100 % Density(per cubic mm)Neonatal Fibroblasts 8000 cellsTGF-β 0.4 pgPDGF 1 pgCollagen 18.75 µgTotal Hyaluronan 80 µgDermal Hyaluronan 80 µgEpidermal Hyaluronan 0 µg
Protocol: one application per week for eight weeks.
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Key Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft TreatmentsSimulation of Apilgraf TreatmentSimulation of Dermagraft Treatment
The Dermagraft Web Site
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Key Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft TreatmentsSimulation of Apilgraf TreatmentSimulation of Dermagraft Treatment
Simulation of Apilgraf Treatment
0 50 100 150 2000
200
400
600
800
1000
time, days
Infla
mm
ator
y M
φ
Treated diabetic wound
Untreated diabetic wound
Normal wound
0 50 100 150 2000
200
400
600
time, days
Rep
air
Mφ
Normalwound
Treated diabeticwound
Untreated diabeticwound
0 50 100 150 2000
500
1000
1500
time, days
Col
lage
n
Untreated diabeticwound
Treated diabeticwound
Normal wound
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Key Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft TreatmentsSimulation of Apilgraf TreatmentSimulation of Dermagraft Treatment
Simulation of Dermagraft Treatment
0 50 100 150 2000
200
400
600
800
1000
time, days
Infla
mm
ator
y M
φ
Treated diabetic wound
Untreated diabetic wound
Normal wound
0 50 100 150 2000
100
200
300
400
500
time, days
Rep
air
Mφ
Normalwound
Treated diabeticwound
Untreated diabeticwound
0 50 100 150 2000
500
1000
1500
time, days
Col
lage
n
Untreated diabeticwound
Treated diabeticwound
Normal wound
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Separating the Treatment ComponentsSeparating Fibroblast FunctionsDemonstration that Hyaluronan Production is Critical
Outline
1 Introduction to Diabetic Wound Healing
2 A Simple Mathematical Model
3 An Expanded Model
4 Using the Model to Understand the Treatments
5 Conclusions
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Separating the Treatment ComponentsSeparating Fibroblast FunctionsDemonstration that Hyaluronan Production is Critical
Separating the Treatment Components
Included: TGFβ, PDGF and collagen
Excluded: hyaluronan, fibroblasts
0 50 100 150 2000
200
400
600
800
1000
time, days
Infla
mm
ator
y M
φ
Untreated diabeticwound
Treated diabeticwound
Normalwound
0 50 100 150 2000
200
400
600
time, days
Rep
air
Mφ
Normal wound
Treated diabeticwound
Untreated diabeticwound
0 50 100 150 2000
500
1000
1500
time, days
Col
lage
n
Untreated diabeticwound
Treated diabeticwound
Normal wound
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Separating the Treatment ComponentsSeparating Fibroblast FunctionsDemonstration that Hyaluronan Production is Critical
Separating the Treatment Components
Included: fibroblasts
Excluded: TGFβ, PDGF, collagen and hyaluronan
0 50 100 150 2000
200
400
600
800
1000
time, days
Infla
mm
ator
y M
φ
Untreated diabeticwound
Treated diabeticwound
Normalwound
0 50 100 150 2000
200
400
600
time, days
Rep
air
Mφ
Normal wound
Treated diabeticwound
Untreated diabeticwound
0 50 100 150 2000
500
1000
1500
time, days
Col
lage
n
Untreated diabeticwound
Treated diabeticwound
Normal wound
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Separating the Treatment ComponentsSeparating Fibroblast FunctionsDemonstration that Hyaluronan Production is Critical
Separating Fibroblast Functions
Fibroblasts secrete a range of regulatory chemicals:
TGFβ
PDGF
collagen
collagenase
hyaluronan
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Separating the Treatment ComponentsSeparating Fibroblast FunctionsDemonstration that Hyaluronan Production is Critical
Separating Fibroblast Functions
Fibroblasts secrete a range of regulatory chemicals:
TGFβ
PDGF
collagen
collagenase
hyaluronan
Question: which of these are critical to treatment success?
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Separating the Treatment ComponentsSeparating Fibroblast FunctionsDemonstration that Hyaluronan Production is Critical
Separating Fibroblast Functions
Fibroblasts secrete a range of regulatory chemicals:
TGFβ
PDGF
collagen
collagenase
hyaluronan
Question: which of these are critical to treatment success?
Answer: our model predicts that only hyaluronan secretionis critical
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
Separating the Treatment ComponentsSeparating Fibroblast FunctionsDemonstration that Hyaluronan Production is Critical
Demonstration that Hyaluronan Production is Critical
Using the model, we simulate treatment with fibroblasts thathave been modified to not produce hyaluronan, but areotherwise normal.
0 50 100 150 2000
200
400
600
800
1000
time, days
infla
mm
ator
y M
φ
Normalwound
Untreated diabetic wound
Modified fibroblast treated diabetic wound
0 50 100 150 2000
100
200
300
400
500
time, days
repa
ir M
φ
Modified fibroblast treated diabetic wound
Untreated diabetic wound
Normal wound
0 50 100 150 2000
500
1000
1500
time, days
colla
gen,
ug
Untreated diabetic wound
Modified fibroblast treated diabetic wound
Normal wound
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
ConclusionsPredictionsReferences
Outline
1 Introduction to Diabetic Wound Healing
2 A Simple Mathematical Model
3 An Expanded Model
4 Using the Model to Understand the Treatments
5 Conclusions
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
ConclusionsPredictionsReferences
Conclusions
The key component ofApligraf and Dermagrafttreatments is theproduction of hyaluronanby the added fibroblasts
Hyaluronan initiateshealing by switchingmacrophages from theinflammatory to therepair phenotype
� � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � α 1−α
� � � � ! � ! � � � � � � � � � � � � � � � � � � � � �" � � � � � � � � � � � � � � �Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
ConclusionsPredictionsReferences
Predictions
1 For both therapies, the current protocols could be improved2 Diabetic wound healing can be initiated simply by the
addition of hyaluronan
0 50 100 150 2000
200
400
600
800
1000
time, days
Infla
mm
ator
y M
φ
Untreated diabeticwound
Treated diabeticwound
Normalwound
0 50 100 150 2000
200
400
600
time, days
Rep
air
Mφ
Normal wound
Treated diabeticwound
Untreated diabeticwound
0 50 100 150 2000
500
1000
1500
time, days
Col
lage
n
Untreated diabeticwound
Treated diabeticwound
Normal wound
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
ConclusionsPredictionsReferences
The Hyalofill Web Site
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
ConclusionsPredictionsReferences
References
H.V. Waugh, J.A. Sherratt.Macrophage dynamics in diabetic wound healing.Bulletin of Mathematical Biology 68, 197-207 (2006)
H.V. Waugh, J.A. Sherratt.Modelling the effects of treating diabetic wounds withengineered skin substitutes.Wound Repair and Regeneration 15, 556-565 (2007)
These papers are available from my web site:www.ma.hw.ac.uk/∼jas
Both the basic and extended models have been translated intoCellML by the University of Auckland; see www.cellml.org
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing
Introduction to Diabetic Wound HealingA Simple Mathematical Model
An Expanded ModelUsing the Model to Understand the Treatments
Conclusions
ConclusionsPredictionsReferences
List of Frames
1 Introduction to Diabetic Wound Healing
Diabetic Wound Healing
Why is Wound Healing Impaired in Diabetics?
2 A Simple Mathematical ModelModel Ingredients
Model EquationsTypical Model SolutionSteady States and Model Predictions
3 An Expanded ModelKey Interactions in the Expanded ModelTypical Solution of the Expanded ModelApligraf and Dermagraft Treatments
Simulation of Apilgraf Treatment
Simulation of Dermagraft Treatment
4 Using the Model to Understand the Treatments
Separating the Treatment Components
Separating Fibroblast Functions
Demonstration that Hyaluronan Production is Critical
5 ConclusionsConclusionsPredictionsReferences
Jonathan Sherratt www.ma.hw.ac.uk/∼jas Macrophage dynamics in diabetic wound healing