steven r mcdougall institute of petroleum engineering heriot-watt university edinburgh scotland...
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Steven R McDougallInstitute of Petroleum Engineering
Heriot-Watt University
Edinburgh
Scotland
Mathematical Modelling of Dynamic Adaptive
Tumour-Induced Angiogenesis
Tumour Growth and Angiogenesis, 11-19 January 2006, Hsinchu, Taiwan
www.pet.hw.ac.uk 2
Co-Investigators
• M A J Chaplain Dept. of MathematicsUniversity of Dundee
• A R A Anderson Dept of Mathematics, University of Dundee
• A Stéphanou Institute of Petroleum Engineering,
Heriot-Watt University
www.pet.hw.ac.uk 3
Outline• Background
• Modelling Tumour-Induced Vasculature
– Continuum scale formulation of growth
– Discrete (capillary) scale growth
– Improved continuous-discrete coupling through MMP
• Incorporating Flow
– Flow in porous media
– Blood flow and drug delivery
• 3D Dynamic Adaptive Tumour-Induced Angiogenesis
– Vascular remodelling and evolving bed architecture
• Conclusions and future work
www.pet.hw.ac.uk 4
Petroleum Engineering and Medicine??
• Multiphase flow at the micro-scale has been investigated over the past 15 years at Heriot-Watt University
• Software resulting from this research is widely used by the oil industry to interpret laboratory experiments and reservoir behaviour
• However, the underlying modelling framework can also be adapted to examine a wide range of problems in the clinical arena
• We will focus here upon a particular application in clinical oncology – tumour-induced angiogenesis
www.pet.hw.ac.uk 5
Modelling Rationale
• Use mathematical modelling techniques to give some insight into the underlying physical and biological processes governing tumour-induced angiogenesis
• Reproduce experimental observations and suggest additional laboratory studies
• Use insights gained from modelling to suggest new treatment strategies and scheduling protocols
Modelling Tumour-Induced Vasculature
www.pet.hw.ac.uk 7
3 Phases of Solid Tumour Growth
• Early avascular phase– tumour spheroid 106 cells– max diameter ~ 2mm– necrotic core– thin proliferating rim
• Angiogenesis and vascularisation– capillary network formation– blood supply to tumour– additional growth
• Invasion and metastasis– attachment to BM– collagenase production– addressins at distant sites
www.pet.hw.ac.uk 8
• Hypoxic tumour cells secrete tumour angiogenic factors (TAFs)
• Endothelial cells degrade their basement membrane
• These migrate towards the tumour
• Capillary branching increases towards the tumour
• Tumour penetrated and nutrient supply begins
Tumour-Induced Angiogenesis
www.pet.hw.ac.uk 9
Vascular Growth SchematicParent Vessel
Tumour Surface
TAF
FN
TISSUE
www.pet.hw.ac.uk 10
Vascular Growth - Continuum Approach
f(x,f(x, y,y, t)t)tt
==
n(x,n(x, y,y,t)t)
tt == DD n n 22 (n(nf )f )
n fn f
c(x,c(x, y,y,t)t)
tt ==
ncnc
(n(nc )c )
nn
c (x,y,t) ~ c (x,y,t) ~ TAF ConcentrationTAF Concentration
f (x,y,t) ~ f (x,y,t) ~ Matrix Macromolecule (fibronectin)Matrix Macromolecule (fibronectin)
n (x,y,t) ~ n (x,y,t) ~ Endothelial Cell DensityEndothelial Cell Density
www.pet.hw.ac.uk 11
EC Density Profile
www.pet.hw.ac.uk 12
• Use discretised form of continuum equations to migrate capillary sprouts and grow capillaries
• Capillary branching, anastomosis and cell mitosis all included in the model
• 2 tumour geometries considered
– linear TAF source
– circular TAF source
Discrete Model
www.pet.hw.ac.uk 13
Discrete Model
l,l, mmqq++11
l,l, mmqq
00 ll++ 1,1, mmqq
11 ll-1,-1, mmqq
22 l,l, mm + + 11qq
33 l,l, mm -1-1qq
44
l,l, mmqq++11
l,l, mmqq
nn == nn PP nn PP nn PP nn PP nn PP
cc == cc
ff == ff l,l, mmqq++11
l,l, mmqq kk nn
l,l, mmqq
1 1 kk l,l, mmqqnn
1 1
withwith x=lhx=lh, , y=mhy=mh andand t=qkt=qk
00 PP
33 PP
22 PP
44 PP
11 PP
kk nnl,l, mmqq
www.pet.hw.ac.uk 14
Movement Weighting
11 22 22
22 22 22
33 22 22
44 22 22
PP ==k Dk Dhh
kk
44hh
PP == k Dk Dhh
++kk
44hh
PP == k Dk Dhh
kk
44hh
PP == k Dk Dhh
++ kk
44hh
) ) (( ll++1,1,mmqq
ll-1,-1,mmqqcc cc
l,l,mm++11qqff ff l,l,mm-1-1
qq(( ))
) ) (( ll++1,1,mmqq
ll-1,-1,mmqqff ff++
) ) (( ll++1,1,mmqq
ll-1,-1,mmqqcc cc ) ) (( ll++1,1,mm
qqll-1,-1,mmqqff ff++
l,l,mm++11qqcc cc l,l,mm-1-1
qq(( )) ++
l,l,mm++11qqff ff l,l,mm-1-1
qq(( )) l,l,mm++11qqcc cc l,l,mm-1-1
qq(( )) ++
00PP == 1 1 44 k Dk Dhh
22
ll++1,1,mmqq
ll-1,-1,mmqq
l,l,mmqq
l,l,mm++11qq
l,l,mm-1-1qq kk
hh(( cc ++ cc -- 44cc ++ cc ++ cc )) 22
ll++1,1,mmqq
ll-1,-1,mmqq
l,l,mmqq
l,l,mm++11qq
l,l,mm-1-1qq kk
hh(( ff ++ ff -- 44ff ++ ff ++ ff ))22
www.pet.hw.ac.uk 15
Numerical Simulation
• At each time step
– Solve fibronectin and TAF equations
– Generate 5 directional coefficients (7 in 3D) P0 to P4
– Compute probability ranges R0 to R4
– Generate random number in (0,1)
– Determine appropriate growth direction
www.pet.hw.ac.uk 16
Branching and Anastamosis
Sprout SproutSprout
Loop formed by anastomosis
Branching at sprout tipto form two new sprouts
Parent Vessel
Sprout tip
www.pet.hw.ac.uk 17
Sample Results
Linear Source Circular Source
www.pet.hw.ac.uk 18
• Matrix metalloproteinases explicitly included
• MMP produced locally by individual ECs
• MMP also diffuses and decays
• n equation now only used to extract directional coefficients
,)(2 haptotaxischemotaxisrandom
fncncnDt
n
.
,
,
2 mmnt
m
mfnt
f
cnt
c
i
i
i
Improved Continuum-Discrete Coupling
www.pet.hw.ac.uk 19
Simulation Results – Large Tumour
Tumour Surface Tumour Surface
Parent Vessel Parent Vessel
Capillary Network Enzyme Concentrations
www.pet.hw.ac.uk 20
Tumour Surface Tumour Surface
Parent Vessel Parent Vessel
Capillary Network Enzyme Concentrations
Simulation Results – Small Tumour
www.pet.hw.ac.uk 21
Comparison with Experiment
Experiment Simulation
www.pet.hw.ac.uk 22
Experiment Simulation
Comparison with Experiment
Incorporating Flow
www.pet.hw.ac.uk 24
a
N S
ShaleAmalgamation surfaceOutcrop termination
100 m
Flow Through Porous Media
www.pet.hw.ac.uk 25
Flow Through Porous Media
(q)i . t = (VbS)i
Qo
Qw
Qo
Qw
Vb, ,So, Sw
www.pet.hw.ac.uk 26
Network Model
www.pet.hw.ac.uk 27
Observing the PhysicsPressure depletion in a reservoir
www.pet.hw.ac.uk 28
Simulation vs Experiment
Oil displacing water from a water-wet micromodel
www.pet.hw.ac.uk 29
We Have the Fluid Distributions - What About
Flow?Pin Pout
qi=0
qi
Large set of pressure equations
www.pet.hw.ac.uk 30
Viscous-Dominated Gasflood
Isolate each phase and flow
its networkGAS OILOIL
www.pet.hw.ac.uk 31
Apply to Vasculatures
Vasculature 1 - Linear TAF source Vasculature 2 - Circular TAF source
www.pet.hw.ac.uk 32
Chemotherapy Modelling
• Single-phase tracer algorithm developed
• “Chemotherapy drug” at concentration Cmax is injected into the upstream end of the parent vessel
• At each timestep
– total mass of drug flowing into each node calculated
– perfect mixing assumed at nodes
– drug uptake instantaneous within 40m of tumour surface
www.pet.hw.ac.uk 33
Initial Assumptions
• Rigid, impermeable cylinders used for capillaries
• Blood treated as Newtonian fluid (constant viscosity)
– no correlation with haematocrit
• No reaction kinetics included in uptake function
• Vascular network is 2D
www.pet.hw.ac.uk 34
Input Data• Four suites of simulations performed
– 2 vasculatures x continuous infusion
– 2 vasculatures x 30s bolus injection
• 5 sets of input data for each suite (0<t<2500s)
Base Run Run 1 Run 2 Run 3 Run 4
blood (Pa.s) 4 x 10-3 1 x 10-3 8 x 10-3 4 x 10-3 4 x 10-3
P (Pa) 800 800 800 800 800Rcap (m) (4, 4.01) (4, 4.01) (4, 4.01) (2, 2.01) (3, 3.01)Rart (m) 10 10 10 10 10
www.pet.hw.ac.uk 35
Continuous Infusion into Vasculature 1
Vasculature 1
www.pet.hw.ac.uk 36
Effect of Blood ViscosityMass in Vasculature
0
2
4
6
8
10
12
0 5 10 15 20 25 30
t*
M*
M* 1cP <r>=4
M* 4cP <r>=4
M* 8cP <r>=4
Uptake in Tumour
0
2
4
6
8
10
12
14
16
18
0 5 10 15 20 25 30
t*
Upt
ake
in T
umou
r
MT* 1cP <r>=4
MT* 4cP <r>=4
MT* 8cP <r>=4
www.pet.hw.ac.uk 37
Effect of Mean Capillary Radius
Mass in Vasculature
0
5
10
15
20
25
30
0 5 10 15 20 25 30
t*
M*
M* 4cP <r>=2
M* 4cP <r>=4
M* 4cP <r>=6
Uptake in Tumour
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20 25 30
t*
Upt
ake
in T
umou
r
MT* 4cP <r>=2
MT* 4cP <r>=4
MT* 4cP <r>=6
www.pet.hw.ac.uk 38
Continuous Infusion into Vasculature 2
Vasculature 2
www.pet.hw.ac.uk 39
Drug Delivery Comparison
Tot Mass in Vasc Circ
0
2
4
6
8
10
12
0 5 10 15 20 25 30
t*
M* M* 4cP <r>=4
Mcirc* 4cP <r>=4
Mass in Tum Circ
0
0.5
1
1.5
2
2.5
3
3.5
0 5 10 15 20 25 30
t*
MT
* MTcirc* 4cP <r>=4
MT* 4cP <r>=4
www.pet.hw.ac.uk 40
Bolus Injection into Vasculature 2
Vasculature 2
www.pet.hw.ac.uk 41
Drug Delivery Comparison
M* Linear vs Circular Bolus
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5 10 15 20 25 30
t*
M* M* 4cP <r>=4 bolus
M* 4cP <r>=4 circ bolus
Uptake in Tum. Lnear vs Circular Bolus
0
0.01
0.02
0.03
0.04
0.05
0.06
0 5 10 15 20 25 30
t*
MT
* MT* 4cP <r>=4 bolus
MT* 4cP <r>=4 circ bolus
www.pet.hw.ac.uk 42
Early Findings
• Bolus injection into a capillary network generated from a circular TAF source
– drug bypasses the tumour completely
• Important implications for chemotherapy strategies
– structure of associated vasculature should be considered when planning chemotherapy treatments
www.pet.hw.ac.uk 43
Targeting Strategies• 3 a posteriori pruning
algorithms considered
– random
– low-flow
– bottlenecks
• Motivation comes from possible targeting of different areas of vasculature with different cytotoxic compounds
Vessel maturation
www.pet.hw.ac.uk 44
Targeting “High Shear” VesselsDrug targeting high-shear vessels
www.pet.hw.ac.uk 45
However …
www.pet.hw.ac.uk 46
Limitations of Early Model
• Two-dimensional
– is bypassing as significant in 3D?
• Blood treated as a Newtonian fluid
– blood is non-Newtonian and biphasic
– blood viscosity is not constant
• Capillaries treated as rigid and impermeable
– in reality, vasodilation/vasoconstriction occurs
– capillary architecture (i.e. bed topology) is not static
www.pet.hw.ac.uk 47
Additional Limitations• Flow only incorporated after growth has
ended
– in fact, perfusion occurs during growth
• Growth and flow towards a tumour
– what about growth and flow within a tumour
– tumour cords around blood vessels
• Other limitations?
– coupled low-pressure venous system absent
– …….??
www.pet.hw.ac.uk 48
Now begin to relax some of the earlier
assumptions…
www.pet.hw.ac.uk 49
Relaxing Earlier Assumptions
3D Extension
www.pet.hw.ac.uk 50
)1(
d
dPZ th
• Bypassing in 3D not necessarily less than that seen in 2D
• Delivery initially slower in 3D
• Anastomosis density, dimensionality, and duration of infusion all affect uptake
• 97-99% of drug injected bypasses the tumour
• Percolation theory can help here
• Really need Z(y) distribution to understand impact of vascular architecture
Relaxing Earlier Assumptions
www.pet.hw.ac.uk 51
• Blood is non-Newtonian
• Capillaries are dynamic entities
• Dynamic feedback must be addressed
– Blood rheology
– Capillary radius = f1 (shear stress)
– Branching probability = f2 (shear stress)
Relaxing Earlier Assumptions
www.pet.hw.ac.uk 52
(i) Blood is non-Newtonian
Red Blood Cells
Endothelial Cells
22
45.0 1.12
2
1.12
2)()1(1
R
R
R
RHf Drel Blood viscosity depends upon
haematocrit and capillary radius…
1211121115.0
)2(06.017.045.0
)2(101
1
)2(101
11)8.0(
1)45.01(
1)1()(
,44.22.360645.0
RReC
HHf
ee
R
C
CD
D
RR
…parameterised as follows
Relaxing Earlier Assumptions
www.pet.hw.ac.uk 53
),( Dapp HRLHR
PRQ
Dapp ),(8
4
Relaxing Earlier Assumptions
(i) Blood is non-Newtonian
www.pet.hw.ac.uk 54
sD
refmerefwttt
tt
kQH
QkPtRRR
RRR
log)(loglog1
1
Wall shear stress
Transmural pressure
Metabolic stimulus
Shrinking tendency
(ii) Shear stress affects vessel radii
Relaxing Earlier Assumptions
www.pet.hw.ac.uk 55
Haematocrit distribution Radius distribution
R(m)% red cells
HD, R, app are interrelated(Haematocrit/Radius -> Viscosity -> Shear stress -> Radius -> Flow -> Haematocrit. . .)
Relaxing Earlier Assumptions
(ii) Shear stress affects vessel radii
www.pet.hw.ac.uk 56
Sprout Branching
- TAF Concentration
- age of the vessel > thr
Vessel Branching
- magnitude of the WSS
- TAF concentration
- min < age of the vessel < max
(iii) Shear stress affects branching
Relaxing Earlier Assumptions
www.pet.hw.ac.uk 57
Age of the vessels in days
(iii) Shear stress affects branching
Relaxing Earlier Assumptions
www.pet.hw.ac.uk 58
WSS / WSS in Parent Vessel
connection connection
(iii) Shear stress affects branching
Relaxing Earlier Assumptions
www.pet.hw.ac.uk 59
Coupling Growth and Flow
Angiogenesis
(cell migration)
Flow Modelling and
Blood Rheology
Dynamic Adaptive Tumour-Induced
Angiogenesis
www.pet.hw.ac.uk 60
Dynamic Adaptive Tumour-Induced
Angiogenesis(DATIA)
www.pet.hw.ac.uk 61
DATIA• A mathematical model which
simultaneously couples vessel growth with blood flow through the vessels – dynamic adaptive tumour-induced angiogenesis
• Radial adaptations and network remodelling occurs as immediate consequences of primary anastomoses
• Capillary network architectures from the dynamically adaptive model differ radically from those obtained using earlier models.
www.pet.hw.ac.uk 62
Modelling Rationale
• Examine the effects of changing various physical and biological model parameters on the developing vascular architecture
• Simulate chemotherapeutic treatments under different parameter regimes
– identify a number of new therapeutic targets for tumour management.
www.pet.hw.ac.uk 63
Simulation Procedure• Capillaries migrate via the discrete form of the
extended PDE formulation (MMP)
– only tip branching possible initially
– no remodelling without flow
• Vessel branching and remodelling considered only after first anastomoses form
• However
– timescale of EC migration ~ days
– timescale of network perfusion ~ minutes
– so we cannot simply remodel as the migration simulation proceeds
www.pet.hw.ac.uk 64
• Idealised procedure
– model the growth of the capillary network using the endothelial cell migration model;
– pause the migration model whenever a new anastomosis (loop) forms;
– switch timescales (t = MIN(Vcap/Qcap))
– flow/remodel the entire capillary bed using flow model until a new steady-state has been reached (~100s of perfusion);
– resume network growth using the cell migration model on the longer timescale.
• Practical procedure
– flow the network to steady-state at regular intervals during the growth process (determine optimum interval).
Simulation Procedure
www.pet.hw.ac.uk 65
A Posteriori Remodelling
t=0.8 t=2.4 t=3.0
t=4.5 t=72.0 t=300.0
www.pet.hw.ac.uk 66
Full DATIA Simulation
4.0 8.0 8.0+1 12.0
12.0+2 12.0+2+3 30.0
www.pet.hw.ac.uk 67
Increased Sensitivity to max
7.5 8.0 8.5
11.5 12.0
www.pet.hw.ac.uk 68
Additional Sensitivities
P(vb)/2.5 P(vb)/5.0 =0.18 pl*=4.0 HD=0.675
HD=0.225 Pin Pout Pin Pout kp
www.pet.hw.ac.uk 69
Transport Through Adapted Networks
• Use tracer algorithm to quantify the efficiency of different networks in carrying blood-borne material e.g. nutrients, chemotherapy drugs, to the tumour
• Almost all of the drug flows through the dilated backbone
• Poor treatment efficiency
• The architecture of the backbone determines delivery to tumour
t=1.0 t=5.0 t=10.0
t=20.0 t=50.0 t=300.0
www.pet.hw.ac.uk 70
• All masses normalised to the total mass injected into the parent vessel over the course of the simulation.
• Only around 1.5% of the infused tracer-drug even enters the capillary network
• Total mass in the network reaches a plateau after approximately 50s
• It takes another 200-250s before uptake commences.
Transport Through Adapted Networks
0
0.005
0.01
0.015
0 100 200 300 400 500
Time (s)
Mas
s in
Bed
(n
orm
alis
ed)
0.00E+00
2.00E-07
4.00E-07
6.00E-07
0 100 200 300 400 500
Time (s)
Up
take
(n
orm
alis
ed)
www.pet.hw.ac.uk 71
Comparison with Homogeneous Bed
• Uptake values are approximately three orders of magnitude higher that those obtained from the remodelled vasculature
• Highlights the need for incorporating vessel adaptations (dilation/constriction) into any angiogenesis model involving transport issues
0.00E+00
2.00E-04
4.00E-046.00E-04
8.00E-04
1.00E-03
0 100 200 300 400 500
Time (s)
Up
take
(n
orm
alis
ed)
0.00E+00
2.00E-07
4.00E-07
6.00E-07
0 100 200 300 400 500
Time (s)
Up
take
(n
orm
alis
ed)
www.pet.hw.ac.uk 72
Effect of Reducing Haptotaxis• Total mass of tracer-drug
entering the supplying vasculature is almost identical to that observed in the base case simulation (=0.28)
• But drug uptake by the tumour is fifty times greater when lateral migration and vessel branching are reduced
• This suggests that tumours supplied by this type of vasculature would be well-supplied with nutrients and could be expected to grow rapidly
• Paradoxically, such tumours would also be highly susceptible to infused treatments
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 100 200 300 400 500
Time (s)
Mas
s in
Bed
(n
orm
alis
ed)
Rho=0.28
Rho=0.16
0.00E+00
5.00E-06
1.00E-05
1.50E-05
2.00E-05
2.50E-05
3.00E-05
0 100 200 300 400 500
Time (s)
Up
take
(n
orm
alis
ed)
Rho=0.28
Rho=0.16
www.pet.hw.ac.uk 73
Effect of Reducing Haematocrit
• A depressed haematocrit was found to lead to the formation of highly dilated arcades close to the parent vessel
• Could be a possible mechanism for generating vasculatures that are detrimental to tumour growth
• The therapeutic implications of this
– more drug enters the capillary network than entered in the base-case simulation
– but drug delivery to the tumour is reduced by more than three orders of magnitude.
0
2E-11
4E-11
6E-11
8E-11
1E-10
0 100 200 300 400 500
Time (s)
Up
take
(n
orm
alis
ed)
0.00E+00
2.00E-07
4.00E-07
6.00E-07
0 100 200 300 400 500
Time (s)
Up
take
(n
orm
alis
ed)
www.pet.hw.ac.uk 74
Other Pieces of the Puzzle
www.pet.hw.ac.uk 75
Flow Within the Tumour• Tumour
vasculature poorly connected
• Tumour capillary radii larger on average
• Capillaries “leaky”
• Examine flow to and within tumour
• Treatment scheduling
www.pet.hw.ac.uk 76
Flow Within the Tumour
Optimise scheduling to build-up high local drug concentrations
www.pet.hw.ac.uk 77
Examine heterogeneous distributions of nutrient
supply and drug delivery
Transmural Diffusion
Feed oxygen tensions into model => evolving TAF sources
www.pet.hw.ac.uk 78
Conclusions and Future Work
www.pet.hw.ac.uk 79
Conclusions• An extensive theoretical investigation of the process of
tumour-induced angiogenesis has been presented incorporating:
– blood rheological properties
– metabolic constraints
– vessel branching = f(wall shear stress )
• Results from computational simulations have highlighted a number of possible new targets for therapeutic intervention
– manipulating sensitivity to wall shear stress
– haptotactic response of the endothelial cells
– haematocrit
– intravascular pressure
www.pet.hw.ac.uk 80
Conclusions• Explicit coupling of growth and flow leads to
network architectures that differ radically from those found in all previous models.
• Dilated loops (anastomoses) form at an earlier stage close to the parent vessel
– positively reinforcing,
– proximally-dilated capillaries undergo further vessel branching.
– subsequent migration of these additional branches result in high capillary densities in regions distal to the parent vessel
– the number of high-conductivity pathways is consequently greatly reduced close to the tumour surface
www.pet.hw.ac.uk 81
Conclusions• It is apparent from the transport simulations that
highly-dilated loops proximal to the parent vessel remove any possibility of effective treatment via intravenous or intra-arterial infusion.
• However, if a tumour-induced capillary network could be forced to develop in just such a way, by means of some clinical intervention perhaps, then nutrient supply to the tumour could be effectively curtailed.
• The DATIA model provides a useful biomechanical framework within which to examine the possibility of managing high conductance pathways as a means of effectively treating solid tumours
www.pet.hw.ac.uk 82
Future Work• Couple flow model to continuum and CA
tumour models
• Use digitised images of real 2D and 3D vasculatures
• Migration of tumour fragments -> metastases
• Heterogeneous tissues
• Other applications (lymphangiogenesis, therapeutic angiogenesis, wound healing, retinopathy…)
www.pet.hw.ac.uk 83
www.pet.hw.ac.uk 84
Discussion Topics
• Tumour-vessel coupling
– increased tumour pressure
– leaky vessels
• Venous system
– low pressure
• Lymphangiogenesis
– how does this differ from angiogenesis?
www.pet.hw.ac.uk 85
Tumour/Vessel Coupling• Potts model for
tumour integrity?
• Appropriate for breast cancer modelling
– used to examine interface region between disrupted tumour surface and healthy perimeter tissue
www.pet.hw.ac.uk 86
Wound Healing
• Similar techniques are being used to study healing rates in dermal wounds.
• Future work will focus upon the design of dressings that could accelerate the healing process and reduce scarring
www.pet.hw.ac.uk 87
Therapeutic Applications
www.pet.hw.ac.uk 88
Random Targeting
What if we end treatment here?
Broad-based indiscriminate drug
www.pet.hw.ac.uk 89
Targeting Poorly Perfused Vessels
Anti-angiogenic drug targeting immature, poorly perfused vessels
www.pet.hw.ac.uk 90
Outcomes• Random targeting
– Increased flow in distal regions
– Decrease in Z4 and increase in Z3 and Z2
– 130% increase in delivery
– Delivery optimised if we can decrease Z4 proximal to parent vessel but maintain good connectivity close to tumour
• Low-flow targeting
– Flow distribution essentially unchanged
– Treatment accelerated in modified network
www.pet.hw.ac.uk 91
• High shear targeting
– Bottleneck (percolation radius) capillaries removed
– Network shut down effectively after 5% vessels removed
– No delivery of drug but also no delivery of nutrients
• Delivery highly sensitive to network architecture
• Main flowing backbone plays a dual role
– Helps carry treatment to tumour
– Increases bypassing
• Treatments should be aimed at managing this backbone
Outcomes
S R McDougall1 and J A Sherratt2
1 Dept of Petroleum Engineering, Heriot-Watt University
2 Dept of Mathematics, Heriot-Watt University
Discrete Modelling of Collagen Deposition and Alignment During Dermal
Wound Repair
www.pet.hw.ac.uk 93
Dermal Wound
www.pet.hw.ac.uk 94
Collagen Alignment
• Post wounding, fibroblasts respond chemotactically and migrate into wound area
• Collagen deposited
• Fibrin degraded
• Scarring due to collagen alignment
• How can we reduce this?
www.pet.hw.ac.uk 95
Matrix Orientation Model
(After Dallon et al, 1999)
)
)s )
(t
(t(t
dt
t dii
v
vf ) ( f
)
)
(t
(tt t ti
ii
f
fv
) ), ( ( ) 1( ) (f c
) sin() , (
fdt
t dx
)
)
(t
(tt x w ti
i N
ii
f
f
1
) , ( ) , (x f
C
FB
f(x,y)
www.pet.hw.ac.uk 96
Results
Speed=5m/hr Speed=15m/hr
www.pet.hw.ac.uk 97
Tissue Regeneration Model
)
)s( )
(t
(t(t
dt
t dii
v
vf) ,
) (b c
f
)
)
(t
(t
t t
t tti
i
i
i
f
fv
) ), ( (
) ), ( () 1( ) (
f u
f u
) sin() , (
fdt
t dx
) , ( ) , ( ) 1( ) , (t t tx x xb c u
C
b
FB
f(x,y)c
b
)
)
(t
(tt x w ti
i N
ii
f
f
1
) , ( ) , (x f
www.pet.hw.ac.uk 98
Tissue Regeneration Model
),()),((),(
1
twtdpdt
td N
iicc xxc
xc
),(),(),(
1
twtddt
td N
iib xxb
xb
C
b
FB
www.pet.hw.ac.uk 99
Results
www.pet.hw.ac.uk 100
Leukocyte Signalling
212
2
kaka
aD
t
a
aka
aD
t
a12
2
HEALTHY PERIMETER
212
2
kaka
aD
t
a
aka
aD
t
a12
2
www.pet.hw.ac.uk 101
Results
www.pet.hw.ac.uk 102
Future Work
• Clinical applications (TGF isoform issue)
• Wound contraction
• Blood flow and angiogenesis