Cancer can give you Maths

Philip K. MainiCentre for Mathematical Biology

Mathematical Institute;and

Oxford Centre for Integrative Systems Biology,Biochemistry

Oxford

• Very brief overview of cancer growth

• First, mutations lead to cells losing appropriate signalling responses for PROLIFERATION (cell division) and APOPTOSIS (cell suicide)

• Result – a growing mass of cells

mutations

Approx 1mm in diameter

• Nutrient required Hypoxic core TAF (tumour angiogenesis factors)

Avascular tumour Vascular tumour • Invasion Tumour produces proteases – digest ECM• Competition

Normal environment:

Normals

Tumour

Gatenby & Gawlinski Gap

Add H+

),,( 21

1

LNNfN

),( 22

2 LNhLDL

),()(. 21211

2

NNgNNDN

HDF

HHH

2

)( 2

2 dR

dHR

dR

d

R

DVHrr

H

v

T-tumour density

V-vascular density

concHH

Glycolytic pathway

Blood flow removal

Avascular Case: 2RRV vV elsewhere

Nondimensionalise:1rr Necrotic

core2122 '2" rrrrrhhr Proliferation zone,

T = constrrhr 22

Outside tumour

Assume necrosis arises when

constant

Using experimentally determined parameter values necrotic core arises at r = 0.1 cm [avascular case]

Thh

Tumour Growth No normal tissue

•Avascular tumour always reaches a benign steady state

•Vascular tumour is benign if invasive if

3

1

31

32

3

2 )( LRRRSdt

dR

Proliferation

necrotic core

(cf Greenspan 1972)

1Th1Th

ResultsThree regimes of growth:

•If rate of acid removal is insufficient, exponential growth followed by auto-toxicity benign tumourOccurs in avasculars and vasculars if

• vascular tumour displays sustained growth and invades

•Very small tumour – no growth (insufficient acid production to include normal cell death)

1Th1Th

Experimental results (Gatenby)

• PROBLEM – THE GAP PREDICTED BY THIS MODEL IS TOO BIG!!!!!

• Introduce quiescent cells (it is known that excess acid induces quiescence). These cells produce very little acid (Smallbone, Gatenby, PKM in prep)

Metabolic changes during carcinogenesis

K. Smallbone, D.J. Gavaghan (Oxford)R.A. Gatenby, R.J. Gillies (Radiology,

Arizona)J.Theor Biol, 244, 703-713, 2007

Introduction

• Carcinogenesis:– The generation of cancer from normal cells– An evolutionary process: selective pressures promote

proliferation of phenotypes best-suited to their microenvironment

Normal cellsAerobic respiration36 ATP / glucose

Cancer cellsAnaerobic respiration

2 ATP / glucose

Cell-environment Interactions

Nature Rev Cancer 4: 891-899 (2004)

DCIS Model

Model Development

• Hybrid cellular automaton:– Cells as discrete individuals

• Proliferation, death, adaptation

– Oxygen, glucose, H+ as continuous fields– Calculate steady-state metabolite fields after each generation

• Heritable phenotypes:– Hyperplastic: growth away from basement membrane– Glycolytic: increased glucose uptake and utilisation– Acid-resistant: Lower extracellular pH to induce toxicity

Cellular Metabolism

• Aerobic:• Anaerobic:

• Assume:– All glucose and oxygen used in these two processes– Normal cells under normal conditions rely on aerobic respiration

alone

ATP2acidlactic2glucose ATP36CO6O6glucose 22

Two parameters:n = 1/18

1 < k ≤ 500c

cnc

c

kg

g

gh

ga

c

g

:H

)(:ATP

:oxygen

cellglycolytic

cellnormal:glucose

Automaton Rules

• At each generation, an individual cell’s development is governed by its rate of ATP production φa and extracellular acidity h

– Cell death• Lack of ATP:

• High acidity:

– Proliferation

– Adaptation

resistant-acid

normal

T

Ndea h/h

h/hp

)1/()( 00 aap adiv

0aa

Somatic Evolution

P.C. Nowell, The clonal evolution of tumour cell populations, Science, 194 (4260), 23-28 (1976)

Variation in Metabolite Concentrations

glucose

oxygen

H+

t=10, normal epithelium t=100, hyperplasia

t=250, glycolysis t=300, acid-resistance

O2 diffusionlimit

basementmembrane

Typical Automaton Evolution

Cellular evolution was demonstrated. 1 of 3 spheroids in 15 days and 3 of 3 in 30 days demonstrated proliferating clusters of GLUT1

positive clusters of cells in normoxic regions.

• For further details, see Gatenby, Smallbone, PKM, Rose, Averill, Nagle, Worrall and Gillies, Cellular adaptations to hypoxia and acidosis during somatic evolution of breast cancer, British J. of Cancer, 97, 646-653 (2007)

Cancer Growth

Tissue Level Signalling: (Tumour Angiogenesis Factors) Oxygen etc

Cells:Intracellular: Cell cycle,

Molecular elements

Partial Differential EquationsAutomaton Elements

Ordinary differential equations

• Vessels – source of nutrient (oxygen); satisfy Pries-Secomb ??????

• Viscosity – Fahraeus-Linqvist effect

• Cells – to divide or not to divide? Thresholds/cell cycle

• Competition – acid etc

Structural adaptation in normal and cancerous vasculature

(PKM, T. Alarcon, H.M. Byrne, M.R. Owen, J. Murphy)

Blood vessels are not static – they respond to stimuli – mechanical and metabolic. Other stimuli are:

Conducted stimuli: downstream (chemical –ATP? released under hypoxic stress)upstream (along vessel wall – changes in

membrane potential through gap junctions?)

• Model includes the production of VEGF by cells in response to low levels of oxygen (hypoxia). VEGF is an angiogenesis factor – it produces more blood vessels.

Results

• No VEGF production – necrotic cores• VEGF production – extensive hypoxic

regions within the tumour but few necrotic regions

• Downstream signalling – tumours with smaller hypoxic regions, more homogeneous distribution of oxygen

• Upstream signalling – VEGF more concentrated around the hypoxic regions

• Model predicts that the inhomogeneous oxygen concentration leads to lower tumour load but symmetry is broken.

References• Alarcon, Byrne, PKM, JTB, 225, 257-274 (2003) -- inhomogeneous media• Alarcon, Byrne, PKM, Prog. Biophys. And Mol. Biol., 85, 451-472 (2004)• Alarcon, Byrne, PKM, JTB, 229, 395-411 (2004) – cell cycle and hypoxia• Ribba, Alarcon, Marron, PKM, Agur, BMB, 67, 79-99 (2005) – doxorubicin• Alarcon, Byrne, PKM, SIAM J. Mult. Mod. Sim, 3, 440-475 (2005)• Alarcon, Byrne, PKM, Microvascular Research, 69, 156-172 (2005) – design

principles• Byrne, Alarcon, Owen, Webb, PKM, Phil Trans R Soc A, 364, 1563-1578

(2006) --review• Byrne, Owen, Alarcon, Murphy, PKM, Math Models and Methods, 16, 1219-

1241 (2006) – chemotherapy• Betteridge, Owen, Byrne, Alarcon, PKM, Networks and Hetero. Media, 1,

515-535 (2006) -- cell crowding• Alarcon, Owen, Byrne, PKM, Comp and Math Methods in Medicine, 7, 85-

119 (2006) – vessel normalisation

Summary

• Simple model for acid-mediated invasion

• Hybrid model for somatic evolution

• Multiscale model:

effects of heterogeneity

structural adaptation in vessels

drug delivery (NOT COVERED TODAY)

Acknowledgements

• Acid/somatic evolution: Bob Gatenby, Kieran Smallbone, David Gavaghan, Mike Brady, Bob Gillies (Funded – EPSRC DTC)

• Multiscale modelling: Tomas Alarcon, Helen Byrne, Markus Owen, James Murphy, Russel Betteridge (Funded – EU RTN (5th and 6th frameworks) IB, NCI Virtual Tumour)