steven r mcdougall institute of petroleum engineering heriot-watt university edinburgh scotland...

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

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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

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• 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

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Vascular Growth SchematicParent Vessel

Tumour Surface

TAF

FN

TISSUE

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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

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• 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

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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

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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

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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

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Branching and Anastamosis

Sprout SproutSprout

Loop formed by anastomosis

Branching at sprout tipto form two new sprouts

Parent Vessel

Sprout tip

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Sample Results

Linear Source Circular Source

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• 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

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Simulation Results – Large Tumour

Tumour Surface Tumour Surface

Parent Vessel Parent Vessel

Capillary Network Enzyme Concentrations

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Tumour Surface Tumour Surface

Parent Vessel Parent Vessel

Capillary Network Enzyme Concentrations

Simulation Results – Small Tumour

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Comparison with Experiment

Experiment Simulation

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Experiment Simulation

Comparison with Experiment

Incorporating Flow

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a

N S

ShaleAmalgamation surfaceOutcrop termination

100 m

Flow Through Porous Media

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Flow Through Porous Media

(q)i . t = (VbS)i

Qo

Qw

Qo

Qw

Vb, ,So, Sw

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Network Model

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Observing the PhysicsPressure depletion in a reservoir

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Simulation vs Experiment

Oil displacing water from a water-wet micromodel

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We Have the Fluid Distributions - What About

Flow?Pin Pout

qi=0

qi

Large set of pressure equations

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Viscous-Dominated Gasflood

Isolate each phase and flow

its networkGAS OILOIL

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Apply to Vasculatures

Vasculature 1 - Linear TAF source Vasculature 2 - Circular TAF source

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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

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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

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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

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Continuous Infusion into Vasculature 1

Vasculature 1

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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

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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

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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

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Targeting “High Shear” VesselsDrug targeting high-shear vessels

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However …

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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

– …….??

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Now begin to relax some of the earlier

assumptions…

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Relaxing Earlier Assumptions

3D Extension

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)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

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• 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

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(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

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),( Dapp HRLHR

PRQ

Dapp ),(8

4

Relaxing Earlier Assumptions

(i) Blood is non-Newtonian

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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

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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

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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

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Age of the vessels in days

(iii) Shear stress affects branching

Relaxing Earlier Assumptions

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WSS / WSS in Parent Vessel

connection connection

(iii) Shear stress affects branching

Relaxing Earlier Assumptions

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Coupling Growth and Flow

Angiogenesis

(cell migration)

Flow Modelling and

Blood Rheology

Dynamic Adaptive Tumour-Induced

Angiogenesis

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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

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A Posteriori Remodelling

t=0.8 t=2.4 t=3.0

t=4.5 t=72.0 t=300.0

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Full DATIA Simulation

4.0 8.0 8.0+1 12.0

12.0+2 12.0+2+3 30.0

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Increased Sensitivity to max

7.5 8.0 8.5

11.5 12.0

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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

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Flow Within the Tumour• Tumour

vasculature poorly connected

• Tumour capillary radii larger on average

• Capillaries “leaky”

• Examine flow to and within tumour

• Treatment scheduling

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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

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Conclusions and Future Work

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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

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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